Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 1

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 2

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 3

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 4

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 5

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 6

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 7

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 8

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 9

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 10

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 11

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 12

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 13

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 14

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 15

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 16

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 17

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 18

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 19

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 20

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 21

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 22

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 23

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 24

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 25

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 26

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 27

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 28

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 29

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 30

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 31

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 32

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 33

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 34

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 35

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 36

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 37

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 38

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 39

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 40

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 41

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 42

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 43

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

62/87,21���

� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

62/87,21���

� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

62/87,21���

� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

62/87,21���

� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

62/87,21���

� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

62/87,21���

� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

62/87,21���

To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

62/87,21���

To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

62/87,21���

To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

62/87,21���

To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

62/87,21���

eSolutions Manual - Powered by Cognero Page 44

6-3 Elimination Using Addition and Subtraction

Use elimination to solve each system of equations.���5m í p = 7

7m í p = 11

62/87,21���Multiply the second equation by í1 . �

� Then, add this to the first equation. �

� Now, substitute 2 for m in either equation to find the value of p .�

� The solution is (2, 3). Check the solution in both equations. �

�

���8x + 5y = 38 í8x + 2y = 4

62/87,21���Because í8x and 8x have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of x. �

� The solution is (1, 6). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV��

�

���7f + 3g = í6 7f í 2g = í31

62/87,21���Because 7f and 7f ��KDYH�WKH�VDPH�FRHIILFLHQWV��VXEWUDFW�WKH�HTXDWLRQV��

� Now, substitute 5 for g in either equation to find the value of f . �

� The solution is (í3, 5). &KHFN�WKH�VROXWLRQ�LQ�ERWK�HTXDWLRQV�� �

�

���6a í 3b = 27 2a í 3b = 11

62/87,21���Because 3b and 3b have the same coefficients, multiply the second equation by ±1, and add the equations to solve for a�� �

�

� Now, substitute 4 for a in either equation to find the value of b. �

� The solution is (4, í1). Check the solution in each equation. �

�

���CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?

62/87,21���Let x represent one number and y represent the second number.x + y = 24 5x ± y = 12 Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 6 for x in either equation to find the value of y . �

� The two numbers are 6 and 18. Check the numbers in each equation. �

��� �

���5(&<&/,1*� The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they provide 1,275,000 jobs. How many jobs does each industry provide?

62/87,21���Let y represent the number of recycling workers and let x represent the number of waste management workers. x + 1,025,000 = y x + y = 1,275,000 Rearrange the first equation to get the variables on the same side. �

� Because x and x have the same coefficients, subtract the two equations. �

� Now, substitute 1,150,000 for y in either equation to find the value of x. �

� There are 125,000 waste management workers and 1,150,000 recycling workers. Check the solution in each equation. �

��

Use elimination to solve each system of equations.���ív + w = 7

v + w = 1

62/87,21���Because ív and v have opposite coefficients, add the equations. �

� Now, substitute 4 for w in either equation to find the value of v. �

� The solution is (í3, 4). Check the solution in each equation. �

��

���y + z = 4 y í z = 8

62/87,21���Because z and íz have opposite coefficients, add the equations.�

� Now, substitute 6 for y in either equation to find the value of z . �

� The solution is (6, í2). Check the solution in each equation. �

�

���í4x + 5y = 17 4x + 6y = í6

62/87,21���Because 4x and í4x have opposite coefficients, add the equations.�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Check the solution in each equation. �

�

����5m í 2p = 24 3m + 2p = 24

62/87,21���Because 2p and í2p have opposite coefficients, add the equations.�

� Now, substitute 6 for m in either equation to find the value of p . �

� The solution is (6, 3). Check the solution in each equation. �

�

����a + 4b = í4 a + 10b = í16

62/87,21���Because a and a have the same coefficients, subtract the equations.�

� Now, substitute í2 for b in either equation to find the value of a. �

� The solution is (4, í2). Check the solution in each equation. �

�

����6r í 6t = 6 3r í 6t = 15

62/87,21���Because í6t and í6t have the same coefficients, multiply equation 2 by ±1 and then add the equations to solve for r.�

�

� Now, substitute í3 for r in either equation to find the value of t. �

� The solution is (í3, í4). Check the solution in each equation. �

�

����6c í 9d = 111 5c í 9d = 103

62/87,21���Because í9d and í9d have the same coefficients, subtract the equations.�

� Now, substitute 8 for c in either equation to find the value of d. �

� The solution is (8, í7). Check the solution in each equation. �

�

����11f + 14g = 13 11f + 10g = 25

62/87,21���Because 11f and 11f ��KDYH�WKH�VDPH�FRHIILFLHQWV��\RX�FDQ�PXOWLSO\�HTXDWLRQ���E\�í1, then add the equations to find g.�

�

� Now, substitute í3 for g in either equation to find the value of f . �

� The solution is (5, í3). Check the solution in each each equation. �

��� �

����9x + 6y = 78 3x í 6y = í30

62/87,21���Because 6y and í6y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The solution is (4, 7). Check the solution in each equation. �

�

����3j + 4k = 23.5 8j í 4k = 4

62/87,21���Because 4k and í4k have opposite coefficients, add the equations.�

� Now, substitute 2.5 for j in either equation to find the value of k . �

� The solution is (2.5, 4). Check the solution in both equations. �

�

����í3x í 8y = í24 3x í 5y = 4.5

62/87,21���Because í3x and 3x have opposite coefficients, add the equations.�

� Now, substitute 1.5 for y in either equation to find the value of x. �

� The solution is (4, 1.5). Check the solution in each equation. �

�

����6x í 2y = 1 10x í 2y = 5

62/87,21���Because í2y and í2y have the same coefficients, subtract the equations.�

� Now, substitute 1 for x in either equation to find the value of y . �

� The solution is (1, 2.5). Check the solution in each equation. �

�

����The sum of two numbers is 22, and their difference is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 17 for x in either equation to find the value of y .�

The two numbers are 17 and 5.

����Find the two numbers with a sum of 41 and a difference of 9.

62/87,21���Let x represent one number and y represent the second number.

Because y and íy have opposite coefficients, add the equations.�

� Now, substitute 25 for x in either equation to find the value of y .�

� The two numbers are 25 and 16.

����Three times a number minus another number is ��7KH�VXP�RI�WKH�QXPEHUV�LV������)LQG�WKH�QXPEHUV�

62/87,21���Let x represent one number and y represent the second number.

Because íy and y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y . �

� The two numbers are 2 and 9.

����A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?

62/87,21���Let x represent one number and y represent the second number.

Because í2y and 2y have opposite coefficients, add the equations.�

� Now, substitute 4 for x in either equation to find the value of y . �

� The two numbers are 4 and 0.

����72856� The Blackwells and Joneses are going to Hershey¶s Really Big 3D Show in Pennsylvania. Find the adult price and the children¶s price of the show. �

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Because 2x and 2x have the same coefficients, multiply equation 2 by ±1, then add the equations to find y .�

�

� Now, substitute 3.95 for y in either equation to find the value of x. �

� The children¶s tickets are $3.95 and the adult tickets are $5.95.

Use elimination to solve each system of equations.����4(x + 2y) = 8

4x + 4y = 12

62/87,21���Distribute the 4 in the first equation. �

� Because 4x and 4x have the same coefficients, subtract the equations.�

� Now, substitute í1 for y in either equation to find the value of x. �

� The solution is (4, í1).

����3x í 5y = 11 5(x + y) = 5

62/87,21���Distribute the 5 in the second equation.

� Because í5y and 5y have opposite coefficients, add the equations.�

� Now, substitute 2 for x in either equation to find the value of y. �

� The solution is (2, í1).

����4x + 3y = 6 3x + 3y = 7

62/87,21���Because 3y and 3y have the same coefficients, multiply equation 2 by í1, and add the equations to find x..�

�

� Now, substitute í1 for x in either equation to find the value of y. �

�

The solution is .

����6x í 7y = í26 6x + 5y = 10

62/87,21���Because 6x and 6x have the same coefficients, subtract the equations.�

� Now, substitute 3 for y in either equation to find the value of x. �

�

The solution is .

����

62/87,21���

Because y and í y have the same coefficients, add the equations.

�

� Now, substitute 12 for x in either equation to find the value of y. �

� The solution is .

����

62/87,21���

Because x and í x have opposite coefficients, add the equations.

�

�

Now, substitute �IRU�y in either equation to find the value of x.

�

� The solution is .

����CCS SENSE-0$.,1*� The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. � D�� How tall is the statue? � E�� How tall is the building?

62/87,21���D��

� Because g and ±g have opposite coefficients, add the equations. �

� Now, substitute 311 for b in either equation to find the value of g.�

� The height of the statue is 15.6 ft. � b. The height of the building is 311 ft.

����BIKE 5$&,1*� Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams. � D�� Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S. teams and non-U.S. teams. � E�� Use elimination to find the solution of the system of equations. � F�� Interpret the solution in the context of the situation. � G�� Graph the system of equations to check your solution.

62/87,21���a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y. � �

� E�� Rearrange the second equation to get the variables on the same side. �

� Because y and ±y have opposite coefficients, add the equations. �

� Now, substitute 48 for x in either equation to find the value of y. �

(48, 18) � F�� This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.� d.

����6+233,1*� Let x represent the number of years since 2004 and y represent the number of catalogs. �

� D�� Write a system of equations to represent this situation. � E�� Use elimination to find the solution to the system of equations. � F�� Analyze the solution in terms of the situation. Determine the reasonableness of the solution.

62/87,21���a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and the growth rates represent the slopes. Write the equation for number of each type of catalogs. RQOLQH���y = 1293x + 7440 SULQW�����y = ±1364x + 3805 � b. Since both equations contain y, use elimination. Negate the second equation. �

� Now use elimination by addition. �

� Now, substitute í1.36808 for x in either equation to find the value of y. �

� The solution is about (í1.4, 5671.1) � F�� This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems reasonable.

����08/7,3/(�5(35(6(17$7,216� Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.

D���&21&5(7(� Choose a combination of 9 objects and find your score. � E���$1$/<7,&$/� Write and solve a system of equations to find the number of paper clips and pennies used for 15 points. � F���7$%8/$5� Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5. � G���9(5%$/� Does the result in the table match the results in part b? Explain.

62/87,21���D�� 6DPSOH�DQVZHU��,I�\RX�FKRRVH���SHQQLHV�DQG���SDSHU�FOLSV��WKH�VFRUH�ZLOO�EH����������RU���� � E�� Let p represent the number of pennies and c represent the number of paper clips. p + c = 9 3p + c = 15 Since both equations contain c, use elimination by subtraction. �

� Now, substitute 3 for p in either equation to find the value of c. �

� So, the solution is p = 3, c = 6. � F�� Sample answer:

� G�� Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no other way of obtaining a score of 15 points.

����5($621,1*� Describe the solution of a system of equations if after you added two equations the result was 0 = 0.

62/87,21���If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is consistent and dependent. The two equations represent the same line. For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely many solutions. When you write each equation in slope-intercept IRUP��\RX�JHW�� ���6LQFH�ERWK�HTXDWLRQV�KDYH�WKH�VDPH�VORSH-intercept form, the equations represent the same line.

����5($621,1*� What is the solution of a system of equations if the sum of the equations is 0 = 2?

62/87,21���The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel lines. For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.��Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have no points in common. So, there is QR�VROXWLRQ���

����23(1�(1'('� Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such systems.

62/87,21���Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a í b = 4.; a system that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).

����&&66�6758&785(� The solution of a system of equations is (í3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system. Explain how you derived this equation.

62/87,21���Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system FRXOG�EH��íx + y = 5.; I used the solution to create another equation with the coefficient of the x-term being opposite of its corresponding coefficient.

����&+$//(1*(� The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is ±4. Define the variables and write the system of equations that you would use to find the number. Then solve the system and find the number.

62/87,21���Let a = the tens digit of the number, and let b = the ones digit of the number.a ± b = ±4; a + b = 8 6ROYH�XVLQJ�HOLPLQDWLRQ�� �

� Find b�� �

� Then the number is 26.

����:5,7,1*�,1�0$7+� Describe when it would be most beneficial to use elimination to solve a system of equations.

62/87,21���Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite FRHIILFLHQWV�LQ�WKH�HTXDWLRQV�� If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps. �

� It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides RI�WKH�HTXDWLRQV�� Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14. �

� 8VLQJ�HOLPLQDWLRQ�ZRXOG�ILUVW�UHTXLUH�UHZULWLQJ�WKH�HTXDWLRQV�VR�WKDW�WKH�YDULDEOHV�ZRXOG�DOLJQ��

� 7KHQ�WKH�VROXWLRQ�FRXOG�EH�IRXQG�E\�XVLQJ�HOLPLQDWLRQ�E\�VXEWUDFWLRQ��

� Substitution might be a more beneficial method to use in this case.

����6+257�5(63216(� Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. Ifthe train has been traveling for an hour, how many more hours is her train ride?

62/87,21���Distance = rate ջ time �

� It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.

����*(20(75<� Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area? � $� A = w � %� V = Bh � &� P = 2 + 2w � '� c2 = a2 + b2

62/87,21���Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the Pythagorean Theorem.

����If the pattern continues, what is the 8th number in the sequence?

�

)��

�

*��

�

+��

�

-��

62/87,21���If the pattern continues, the following numbers will be in the series:

.

This means that choice F is correct.

����What is the solution of this system of equations? � x + 4y = 1 2x í 3y = í9 � $� (2, í8) � %� (í3, 1) � &� no solution � '� infinitely many solutions

62/87,21���Solve the first equation for x since its coefficient is 1. �

� Substitute 1 - 4y for x in the second equation to find the value of y .�

� Now, substitute 1 for y in either equation to find the value of x. �

� The solution is (í3, 1). Therefore, the correct choice is B.

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

����y = 6x 2x + 3y = 40

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� Now, substitute 2 for x in either equation to find the value of y.�

� The solution is (2, 12).

����x = 3y 2x + 3y = 45

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� Now, substitute 5 for y in either equation to find the value of x.�

� The solution is (15, 5).

����x = 5y + 6 x = 3y í 2

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� Now, substitute í4 for y in either equation to find the value of x.�

� The solution is (í14, í4).

����y = 3x + 2 y = 4x í 1

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� Now, substitute 3 for x in either equation to find the value of y. �

� The solution is (3, 11).

����3c = 4d + 2 c = d í 1

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� Now, substitute í5 for d in either equation to find the value of c. �

� The solution is (í6, í5).

����z = v + 4 2z í v = 6

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� Now, substitute í2 for v in either equation to find the value of z.�

� The solution is (í2, 2).

����),1$1&,$/�/,7(5$&<� Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has $26 and plans to save $13 per week. D�� In how many weeks will Gregorio and Javier have saved the same amount of money? E�� How much will each person have saved at that time?

62/87,21���a. Let w represent the number of weeks and t represent the total savings. So the two equations are:Gregario: y = 35 + 10w��� Javier: y = 26 + 13w Substitute 35 + 10w for t in the second equation to find the value of w. �

� So, They will have the same amount of money saved in 3 weeks. � b. Substitute 3 for w in either equation. �

� Therefore, Gregorio and Javier will each have saved $65.

����*(20(75<� A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.

62/87,21���Use the equations of the lines to find the slopes of the sides of the parallelogram. �

lies on the vertical line described by x = ±4, so its slope is undefined. lies on the vertical line described by x = 3, so its slope is undefined. lies on the line described by , so it has a slope of .

lies on the line described by , so it has a slope of . � Since each pair of opposite sides has the same slope, the opposite sides are parallel.A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Therefore, ABCD is a parallelogram.

Solve each equation. Check your solution.����6u = í48

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To check this answer, substitute í8 into the original equation. 6(í8) = í48, so the solution checks.

����75 = í15p

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To check this answer, substitute í5 into the original equation. í15(í5) = 75, so the solution checks.

����

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To check this answer, substitute 12 into the original equation. = 8, so the solution checks.

����

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To check this answer, substitute í20 into the original equation. = 15, so the solution checks.

Simplify each expression. If not possible, write simplified.����6q í 3 + 7q + 1

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����7w2 í 9w + 4w2

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����10(2 + r) + 3r

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����5y í 7(y + 5)

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eSolutions Manual - Powered by Cognero Page 45

6-3 Elimination Using Addition and Subtraction

����7w2 í 9w + 4w2

62/87,21���

����10(2 + r) + 3r

62/87,21���

����5y í 7(y + 5)

62/87,21���

eSolutions Manual - Powered by Cognero Page 46

6-3 Elimination Using Addition and Subtraction