Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
Name Class Date
3-1 Review
As you solve a system of equations, remember the following ideas.
• Lines that have the same slopes but different y-intercepts are parallel and will never intersect. These systems are inconsistent.
• Lines that have both the same slope and the same y-intercept are the same line and will intersect at every point. These systems are dependent.
• Lines that have different slopes will intersect, and the system will have one solution. These systems are independent.
Using a graph or a table, what is the solution of the system of equations?
y = −2x + 8 Write both equations in y = mx + b form.
y = x + 2
y = –2x + 8
Graph the line y = −2x + 8. Graph the line y = x + 2. Circle the point of intersection.
y = x + 2
x = 2, y = 4 Determine the x- and y-coordinates of the point of intersection.
The solution is the ordered pair (2, 4).
Check 2(2) + 4 8 Check by substituting the solution into both equations. 4 + 4 8
8 = 8 4 − 2 2
2 = 2 Exercises Solve each system by graphing or using a table. Check your answers.
1. 2. 3.
4. 5. 6.
7. Which point lies on both Line 1 and Line 2?
(0, 0) (1.875, 1.875)
(2.05, 2.05) (2, 2)
Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
9
Solving Systems Using Tables and Graphs
Name Class Date
3-1 Review (continued)
The table shows the winning times for the Olympic 400-M dash. Use your graphing calculator to find linear models for women’s and men’s winning times. Assuming the trends in the table continue, when will the women’s winning time and the men’s winning time be equal? What will that winning time be?
SOURCE: International Olympic Committee
Step 1 Enter the data into lists on your calculator. L1: number of years since 1968 (value for x) L2: men’s winning times in seconds (value for y1) L3: women’s winning times in seconds (value for y2)
Step 2 Use LinReg(ax + b) to find linear models. This determines the equation of the lines of best f t for the selected data. Use L1 and L2 for the men’s winning times. Use L1 and L3 for the women’s winning times.
Step 3 Graph each model. Use the Intersect feature on the graphing calculator to find the solution of the system. The solution is x = 99.72093 and y = 42.00168.
The linear model shows that if the table’s trends continue, the times for men and women will be equal about 100 years after 1968, in 2068. The winning time will be about 42 seconds.
Exercise 8. The table shows the winning times for Olympic 500-M speed skating. Assuming these trends
continue, when will the women’s winning time equal the men’s winning time? What will that winning time be?
SOURCE: International Olympic Committee
Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
10
Solving Systems Using Tables and Graphs
Winning Times for the Olympic 400-M Dash (seconds) Year 1968 1972 1976 1980 1984 1988 1992 1996 2000
Men’s Times Women’s Times
43.86 44.66 44.26 44.60 44.27 43.87 43.50 43.49 43.84
52.03 51.08 49.29 48.88 48.83 48.65 48.83 48.25 49.11
Winning Times for the Olympic 500-M Speed Skating (seconds) Year 1968 1972 1976 1980 1984 1988 1992 1994 1998 Men’s Times Women’s Times
40.30 39.44 39.17 38.03 38.19 36.45 37.14 36.33 35.59
46.10 43.33 42.76 41.78 41.02 39.10 40.33 39.25 38.21
Name Class Date
3-2 Review
Follow these steps when solving by substitution.
Step 1 Solve one equation for one of the variables.
Step 2 Substitute the expression for this first variable into the other equation. Solve for the second variable.
Step 3 Substitute the second variable’s value into either equation. Solve for the first variable.
Step 4 Check the solution in the other original equation.
What is the solution of the system of equations? 4 + 3 = 10
+ 2 = 10x yx y
⎧⎨⎩
Step 1 x = −2y + 10 Solve one equation for x.
Step 2 4(−2y + 10) + 3y = 10 Substitute the expression for x into the other equation. −8y + 40 + 3y = 10 Distribute.
–5y = −30 Combine like terms. y = 6 Solve for y.
Step 3 x + 2(6) = 10 Substitute the y value into either equation. x + 12 = 10 Simplify.
x = –2 Solve for x.
Step 4 4(−2) + 3(6) 0 10 Check the solution in the other equation. −8 + 18 10 Simplify. 10 = 10
The solution is (−2, 6).
Exercises Solve each system by substitution.
1. 2. 3. 4.
Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
19
Solving Systems Algebraically
Name Class Date
3-2
Review (continued)
Follow these steps when solving by elimination.
Step 1 Arrange the equations with like terms in columns. Circle the like terms for which you want to obtain coefficients that are opposites.
Step 2 Multiply each term of one or both equations by an appropriate number.
Step 3 Add the equations. Step 4 Solve for the remaining variable.
Step 5 Substitute the value obtained in step 4 into either of the original equations, and solve for the other variable.
Step 6 Check the solution in the other original equation.
What is the solution of the system of equations? 2 – 5 = 113 – 2 = –12x yx y
⎧⎨⎩
Step 1 + 5y = 11 Circle the terms that you want to make opposite. − 2y = −12
Step 2 6x + 15y = 33 Multiply each term of the first equation by 3. −6x + 4y = 24 Multiply each term of the second equation by −2.
Step 3 19y = 57 Add the equations. Step 4 y = 3 Solve for the remaining variable.
Step 5 3x − 2(3) = −12 Substitute 3 for y to solve for x. x = −2
Step 6 2(−2) + 5(3) 11 Check using the other equation. −4 + 15 11
11 = 11
The solution is (−2, 3). You can also check the solution by using a graphing calculator.
Exercises Solve each system by elimination.
5. 6. 7. 8.
9. Reasoning Why does a system with no solution represent parallel lines?
Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
20
Solving Systems Algebraically
Name Class Date
3-3 Review
Solving a System by Using a Table
An English class has 4 computers for at most 18 students. Students can either use the computers in groups to research Shakespeare or to watch an online performance of Macbeth. Each research group must have 4 students and each performance group must have 5 students. In how many ways can you set up the computer groups?
Step 1 Relate the unknowns and define them with variables.
x = number of research groups, y = number of performance groups
number of research groups + number of performance groups ≤ 4
4. number of research groups + 5 number of performance groups ≤ 18
Step 2 Make a table of values for x and y that satisfy the first
inequality. The replacement values for x and y must be whole numbers.
Step 3 In the table, check each pair of values to see which satisfy the other inequality. Highlight these pairs. These are the solutions of the system.
You can have: 0 groups doing research and 0, 1, 2, or 3 groups watching performances or 1 group doing research and 0, 1, or 2 groups watching performances or 2 groups doing research and 0, 1, or 2 groups watching performances or 3 groups doing research and 0 or 1 group watching performances or 4 groups doing research and 0 groups watching performances
Exercises Find the whole number solutions of each system using tables.
1. 2. 3. 4.
Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
29
Systems of Inequalities
x y 0 4, 3, 2, 1, 0
1 3, 2, 1, 0
2 2, 1, 0 3 1, 0
4 0
x + y ≤ 4 4x + 5y ≤ 18
x y 0 4, 3,2, 1, 0
1 3, 2,1, 0
2 2, 1,0
3 1, 0 4 0
Name Class Date
3-3
Review (continued)
Solving a System by Graphing
What is the solution of the system of inequalities?
Step 1 Solve each inequality for y. 2x − y > 1 x + y ≥ 3 −y > −2x + 1 and y ≥ −x + 3 y < 2x − 1
Step 2 Graph the boundary lines. Use a solid line for ≥ or ≤ inequalities. Use a dotted line for > and < inequalities.
Step 3 Shade on the appropriate side of each boundary line. The overlap is the solution to the system.
Exercises Solve each system of inequalities by graphing.
5. y ≤ xy ≥ 3x −1⎧⎨⎩
⎫⎬⎭
6. 2 + > 3
– < 2x yx y
⎧⎨⎩
7. > 1 < + 1xy x
⎧⎨⎩
8. + 3 9
2 – > 1x yx y
≤⎧⎨⎩
9. 1 < – – 13
3 + 1
y x
y x
⎧⎪⎨⎪ ≥⎩
10. 4 + 1
+ 2 –1x yx y
≤⎧⎨
≤⎩
11. 3 > 4 – 3yy x≤⎧
⎨⎩
12. 2 + < 33 – < 2x yx y
⎧⎨⎩
13. –2
3 + 2yx y≥⎧
⎨≤⎩
Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
30
Systems of Inequalities
Name Class Date
3-4 Review
What point in the feasible region maximizes P for the objective function P = 10x + 15y? What point minimizes P?
Step 1 Step 2 Step 3
Graph the constraints and Find the coordinates for Evaluate P at each vertex. shade the feasible region. each vertex of the region.
VERTEX P = 10x + 15y
A (0, 0) P = 10(0) + 15(0) = 0
B (16, 0) P = 10(16) + 15(0) = 160
C (12, 4) P = 10(12) + 15(4) = 180
D (0, 10) P = 10(0) + 15(10) = 150
The maximum value of the objective function is 180. It occurs when x = 12 and y = 4. The minimum value of the objective function is 0. It occurs when x = 0 and y = 0.
Exercises Graph each system of constraints. Name all vertices. Then find the values of x and y that maximize or minimize the objective function.
1. 2. 3.
P = 8x + 2y P = x + 3y P = x − 2y
Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
39
Linear Programming
Name Class Date
3-4
Review (continued)
Your school band is selling calendars as a fundraiser. Wall calendars cost $48 per case of 24. You sell them at $7 per calendar. Pocket calendars cost $30 per case of 40. You sell them at $3 per calendar. You make a profit of $120 per case of wall calendars and $90 per case of pocket calendars. If the band can buy no more than 1000 total calendars and spend no more than $1200, how can you maximize your profit if you sell every calendar? What is the maximum profit?
Relate Organize the information in a table.
Define Let x = number of cases of wall calendars
Let y = number of cases of pocket calendars Write Use the information in the table and the definitions of x and y to write the constraints
and the objective function. Simplify the inequalities if necessary.
24x + 40y ≤ 1000 48x + 30y ≤ 1200 23x + 45y ≤ 125 8x + 5y ≤ 200
Objective function: P = 120x + 90y
Step 1 Step 2 Step 3 Graph the constraints and Find the coordinates for Evaluate the objective function shade to see the feasible region each vertex of the region. using the vertex coordinates.
Linear Programming
Wall Calendars Pocket Calendars Total Number of Cases x y
Number of Units 24x 40y 1000
Cost 48x 30y 1200
Profit 120x 90y 120x + 90y
A(0, 0) P = 120(0) + 90(0) = 0
B(25, 0) P = 120(25) + 90(0) = 3000
C(15, 16) P = 120(15) + 90(16) = 3240
D(0, 25) P = 120(0) + 90(25) = 2250
You can maximize your profit by selling 15 cases of wall
calendars and 16 cases of pocket calendars. The maximum profit is $3240.
Exercises 4. Your band decides to sell the wall calendars for $9 each.
a. How many of each type of calendar should you now buy to maximize your profit?
b. What is the maximum profit?
Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
40
Name Class Date
3-5
Review
What is the solution of the system?
6 2 3 9
2 2 9
x y zx y zx y z
+ + =⎧⎪
− + =⎨⎪− + + =⎩
Use elimination. The equations are numbered to make the process easy to follow.
x + y + z = 6 −x + 2y + 2z = 9
3y + 3z = 15
2x − 2y + 3z = 9 −x + 2y + 2z = 9
2x − y + 3z = 9 −2x + 4y + 4z = 18 3y + 7z = 27 3y + 3z = 15 −3y + (−7z) = −27
−4z = −12 z = 3
3y + 3(3) = 15
3y = 6 y = 2
x + 2 + 3 = 6 x = 1
Pair the equations to eliminate x.
Pair a different set of equations.
Multiply equation 3 by 2 to eliminate x. Then add the two equations.
Equations 4 and 5 form a system. Multiply equation 5 by −1 and add to equation 4 to eliminate y and solve for z.
Substitute z = 3 into equation 4 and solve for y.
Substitute the values of y and z into one of the original equations. Solve for x.
The solution is (1, 2, 3).
Exercises Solve each system by elimination. Check your answers.
1. 2. 3.
4. 5. 6.
Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
49
Systems With Three Variables
Name Class Date
What is the solution of the system? Use substitution.
3 2 19 4 2 3 8
3 2 2 15
x y zx y zx y z
+ − =⎧⎪
− + =⎨⎪− + + =⎩
Step 1 Choose an equation that can be solved easily for
one variable. Choose equation 1 and solve for x. x + 3y − 2z = 19
x = −3y + 2z + 19
Step 2 Substitute the expression for x into equations 2 and 3 and simplify.
4(−3y + 2z + 19) − 2y + 3z = 8 −3(−3y + 2z + 19) + 2y + 2z = 15 −12y + 8z + 76 − 2y + 3z = 8 9y − 6z − 57 + 2y + 2z = 15
−14y + 11z = −68 11y − 4z = 72
−56y + 44z = −272 Multiply by4. 121y − 44z = 792 Multiply by11.
65y = 520 y = 8
−14y + 11z = −68 Substitute y=8 into . –14(8) + 11z = −68
11z = 44 z = 4
Step 4 Use one of the original equations to solve for x.
Review_Systems With Three Variables
Step 3 Write the two new equations as a system. Solve for y and z.
−14y +11z = −6811y − 4z = 72⎧⎨⎩
x + 3y − 2z = 19 Substitute y = 8 and z = 4 into x + 3(8) − 2(4) = 19
x = 3
The solution of the system is (3, 8, 4).
Exercises Solve each system by substitution. Check your answers.
7. 4 3 2 7
2 2 3 152 2 6
x y zx y zx y z
− + − =⎧⎪
− + =⎨⎪− + − = −⎩
8. 2 5 5
3 2 7 102 3 6 12
x y zx y zx y z
− − + = −⎧⎪
+ − =⎨⎪− − + = −⎩
9. 2 3 153 4 2 7 2 2 5 22
x y zx y zx y z
− + =⎧⎪
− − =⎨⎪ + + =⎩
10. 2 2 1
2 2 114 3 2 4
x y zx y zx y z
− + − = −⎧⎪
+ + =⎨⎪ − + =⎩
11. 3 2 7
2 5 4 1 4 6 2
x y zx y zx y z
− + =⎧⎪
− − = −⎨⎪ + − = −⎩
12. 3 3 132 4 5 5
5 2 3
x y zx y zx y z
− − = −⎧⎪
+ − = −⎨⎪− + + =⎩
Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
50
Name Class Date
3-6
Review
How can you represent the system of equations with a matrix?
4x − 3y + 5z = −13x + 3y = 32x + 4y + 3z = 17
⎧
⎨⎪
⎩⎪
Step 1 Write each equation in the same variable order. Line up the like variables. Write in variables that have a coefficient of 0.
4 – 3 5 –13 3 0 3
–2 4 3 17
x y zx y zx y z
+ =⎧⎪
+ + =⎨⎪ + + =⎩
Step 2 Write the matrix using the coefficients and constants. Remember to enter a 1 for variables with no numeric coefficient.
4 3 5 –131 3 0 32 4 3 17
⎡ ⎤−⎢ ⎥⎢ ⎥⎢ ⎥−⎣ ⎦
Exercises Write a matrix to represent each system.
1. 2 –33 5x yy+ =⎧
⎨=⎩
2. 3 – 5 2 94 7 32 – 12
x y zx y zx z
+ =⎧⎪
+ + =⎨⎪ =⎩
3. 5 – 3 23 2 64 3 1
x y zy zx y z
+ =⎧⎪
+ =⎨⎪ + + =⎩
4. 2 – 35 4 –5– 2 1
x zy zx y
=⎧⎪
+ =⎨⎪ + =⎩
5. 6
23 – 2 – 5 10
zx yx y z
=⎧⎪
+ =⎨⎪ =⎩
6. 2 – 5 3 4– 2 4 –23 – 2 –5
z x yy x zx z y
+ =⎧⎪
+ + =⎨⎪ + =⎩
Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
59
Solving Systems Using Matrices
Name Class Date
3-6
Review (continued)
What is the solution of the system? 2 5 5– 2 –7x yx y+ =⎧
⎨+ =⎩
Step 1 Write the matrix for the system.
2 5 5
1 2 –7⎡ ⎤⎢ ⎥−⎣ ⎦
Step 2 Multiply Row 2 by 2. Add to Row 1. Replace Row 1 with the sum. Write the new matrix.
2 5 5
+ 2(–1 2 –7)
0 9 –9
Step 3 Divide Row 1 by 9. Write the new matrix.
Solving Systems Using Matrices
0 9 –9
1 2 –7⎡ ⎤⎢ ⎥−⎣ ⎦
0 1 –11 (0 9 –9) 1 2 –79
⎡ ⎤⎢ ⎥−⎣ ⎦
Step 4 Multiply Row 1 by –2. Add to Row 2. Replace Row 2 with the sum. Write the new matrix.
–2(0 1 –1)
+ −1 2 –7
–1 0 –5
Step 5 Multiply Row 2 by 21. Write the new matrix.
−1 −1 0 −5( )
0 11 0
–15
⎡
⎣⎢⎢
⎤
⎦⎥⎥
This matrix is equivalent to the system –1 5yx=⎧
⎨=⎩
. The solution is (5, −1).
Exercises Solve each system of equations using a matrix.
7. 4 3 6– – –1x yx y+ =⎧
⎨=⎩
8. 6 –2– 3 13x yx y+ =⎧
⎨+ =⎩
9. 3 2 –4–4 – 3 7x yx y+ =⎧
⎨=⎩
Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
60
0 1 –1
1 0 –5⎡ ⎤⎢ ⎥−⎣ ⎦
Prentice Hall Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
9
Name Class Date
As you solve a system of equations, remember the following ideas.
• Lines that have the same slopes but diff erent y-intercepts are parallel and will never intersect. Th ese systems are inconsistent.
• Lines that have both the same slope and the same y-intercept are the same line and will intersect at every point. Th ese systems are dependent.
• Lines that have diff erent slopes will intersect, and the system will have one solution. Th ese systems are independent.
Problem
Using a graph or a table, what is the solution of the system of equations? e 2x 1 y 5 8
y 2 x 5 2
y 5 22x 1 8 Write both equations in y 5 mx 1 b form.
y 5 x 1 2
Graph the line y 5 22x 1 8. Graph the line y 5 x 1 2. Circle the point of intersection.
x 5 2, y 5 4 Determine the x- and y -coordinates of the point of intersection.
Th e solution is the ordered pair (2, 4).
Check 2(2) 1 4 0 8 Check by substituting the solution into both equations.
4 1 4 0 8 8 5 8 �
4 2 2 0 22 5 2 �
Exercises
Solve each system by graphing or using a table. Check your answers.
1. e 3x 1 y 5 6
3x 1 y 5 3 2. e22x 1 y 1 3 5 0
x 2 1 5 y 3. e x 1 y 5 3
y 5 3x 2 1
4. e y 5 1 2 x
2x 1 y 5 4 5. e2x 1 2y 5 2
3x 1 2y 5 26 6. e2x 1 y 5 22
22x 1 3y 5 23
7. Which point lies on both Line 1 and Line 2?
(0, 0) (1.875, 1.875)
(2.05, 2.05) (2, 2)
y � �2x � 8
y � x � 2 x
y
O2 4 6 8�2
468
3-1 Reteaching Solving Systems Using Tables and Graphs
xO 2Line 1
4
Line 24
�4
y
(1, 3)
(3, 22)
(2, 1)
(22, 0)
C
(1, 2)
(3, 1)
Prentice Hall Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
10
Name Class Date
Problem
Th e table shows the winning times for the Olympic 400-M dash. Use your graphing calculator to fi nd linear models for women’s and men’s winning times. Assuming the trends in the table continue, when will the women’s winning time and the men’s winning time be equal? What will that winning time be?
Step 1 Enter the data into lists on your calculator.
L1: number of years since 1968 (value for x)
L2: men’s winning times in seconds (value for y1)
L3: women’s winning times in seconds (value for y2)
Step 2 Use LinReg(ax 1 b) to fi nd linear models. Th is determines the equation of the lines of best fi t for the selected data.
Use L1 and L2 for the men’s winning times.
Use L1 and L3 for the women’s winning times.
Step 3 Graph each model. Use the Intersect feature on the graphing calculator to fi nd the solution of the system. Th e solution is x 5 99.72093 and y 5 42.00168.
Th e linear model shows that if the table’s trends continue, the times for men and women will be equal about 100 years after 1968, in 2068. Th e winning time will be about 42 seconds.
Exercise 8. Th e table shows the winning times for Olympic 500-M speed skating.
Assuming these trends continue, when will the women’s winning time equal the men’s winning time? What will that winning time be?
Winning Times for the Olympic 400-M Dash (seconds)
Year 1968
52.03
43.86
51.08
1972
44.66
49.29
1976
44.26
1980
48.88
44.60
1984
48.83
44.27
1988
48.65
43.87
1992
48.83
43.50
1996
48.25
43.49
2000
49.11
43.84Men’sTimes
Women’sTimes
SOURCE: International Olympic Committee
�
�������
Intersectionx � 99.72093 y � 42.00168
Winning Times for the Olympic 500-M Speed Skating (seconds)
Year 1968
46.10
40.30
43.33
1972
39.44
42.76
1976
39.17
1980
41.78
38.03
1984
41.02
38.19
1988
39.10
36.45
1992
40.33
37.14
1994
39.25
36.33
1998
38.21
35.59Men’sTimes
Women’sTimes
SOURCE: International Olympic Committee
3-1 Reteaching (continued)
Solving Systems Using Tables and Graphs
2028; 31.265 seconds
Prentice Hall Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
19
Name Class Date
Follow these steps when solving by substitution.
Step 1 Solve one equation for one of the variables.
Step 2 Substitute the expression for this fi rst variable into the other equation. Solve for the second variable.
Step 3 Substitute the second variable’s value into either equation. Solve for the fi rst variable.
Step 4 Check the solution in the other original equation.
Problem
What is the solution of the system of equations? e 4x 1 3y 5 10
4x 1 2y 5 10
Step 1 x 5 22y 1 10 Solve one equation for x.
Step 2 4(22y 1 10) 1 3y 5 10 Substitute the expression for x into the other equation. 28y 1 40 1 3y 5 10 Distribute. 25y 5 230 Combine like terms. y 5 6 Solve for y.
Step 3 x 1 2(6) 5 10 Substitute the y value into either equation. x 1 12 5 10 Simplify. x 5 22 Solve for x.
Step 4 4(22) 1 3(6) 0 10 Check the solution in the other equation. 28 1 18 0 10 Simplify. 10 5 10 �
Th e solution is (22, 6).
Exercises
Solve each system by substitution.
1. e2x 2 3y 5 2
2x 1 2y 5 5 2. ea 1 3b 5 4
a 5 22 3. e22m 1 n 5 6
27m 1 6n 5 1 4. e 7x 2 3y 5 21
x 1 2y 5 12
3-2 ReteachingSolving Systems Algebraically
x 5 219, y 5 27 a 5 22, b 5 2 m 5 27, n 5 28 x 5 2, y 5 5
Prentice Hall Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
20
Name Class Date
Follow these steps when solving by elimination.
Step 1 Arrange the equations with like terms in columns. Circle the like terms for which you want to obtain coeffi cients that are opposites.
Step 2 Multiply each term of one or both equations by an appropriate number.
Step 3 Add the equations.
Step 4 Solve for the remaining variable.
Step 5 Substitute the value obtained in step 4 into either of the original equations, and solve for the other variable.
Step 6 Check the solution in the other original equation.
Problem
What is the solution of the system of equations? e 2x 1 5y 5 211
3x 2 2y 5 212
Step 1 2x 1 5y 5 211 Circle the terms that you want to make opposite. 3x 2 2y 5 212
Step 2 6x 1 15y 5 33 Multiply each term of the fi rst equation by 3. 26x 1 14y 5 24 Multiply each term of the second equation by 22.
Step 3 19y 5 57 Add the equations.Step 4 y 5 3 Solve for the remaining variable.
Step 5 3x 2 2(3) 5 212 Substitute 3 for y to solve for x. x 5 22
Step 6 2(22) 1 5(3) 0 11 Check using the other equation. 24 1 15 0 11 11 5 11 �
Th e solution is (22, 3). You can also check the solution by using a graphing calculator.
Exercises
Solve each system by elimination.
5. e 3x 1 2y 5 217
3x 2 3y 5 219 6. e25f 1 4m 5 26
22f 2 3m 5 21 7. e23x 2 2y 5 5
26x 1 4y 5 7 8. e22x 2 24y 5 212
10x 1 20y 5 210
9. Reasoning Why does a system with no solution represent parallel lines?
3-2 Reteaching (continued)
Solving Systems Algebraically
If there is no solution, then there are no values of the variables that will make both equations true. This means there is no point that lies on both lines, so the lines never meet and are therefore parallel.
x 5 23, y 5 24 f 5 2, m 5 21 no solution y 5 212x 2 12, where
x is any real number
Prentice Hall Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
29
Name Class Date
Solving a System by Using a Table
Problem
An English class has 4 computers for at most 18 students. Students can either use the computers in groups to research Shakespeare or to watch an online performance of Macbeth. Each research group must have 4 students and each performance group must have 5 students. In how many ways can you set up the computer groups?
Step 1 Relate the unknowns and defi ne them with variables.
x 5 number of research groups, y 5 number of performance groups number of research groups 1 number of performance groups # 4
4 ? number of research groups 1 5 ? number of performance groups # 18
Step 2 Make a table of values for x and y that satisfy the fi rst inequality. Th e replacement values for x and y must be whole numbers.
Step 3 In the table, check each pair of values to see which satisfy the other inequality. Highlight these pairs. Th ese are the solutions of the system.
You can have:0 groups doing research and 0, 1, 2, or 3 groups watching performances or1 group doing research and 0, 1, or 2 groups watching performances or 2 groups doing research and 0, 1, or 2 groups watching performances or3 groups doing research and 0 or 1 group watching performances or4 groups doing research and 0 groups watching performances
Exercises
Find the whole number solutions of each system using tables.
1. e x 1 y , 4
x 1 2y # 10 2. e 6x 2 3y $ 1
6x 1 3y # 21 3. e x 1 y $ 5
y , 22x 1 8 4. e y , 3
4x 1 2y , 12
x 1 y # 44x 1 5y # 18
3-3 ReteachingSystems of Inequalities
x
0 4, 3, 2, 1, 0
1 3, 2, 1, 0
2 2, 1, 0
3 1, 0
4 0
y
x
0 4, 3, 2, 1, 0
1 3, 2, 1, 0
2 2, 1, 0
3 1, 0
4 0
y
(0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (3, 0)
(1, 0), (2, 0), (2, 1), (3, 0), (3, 1)
(0, 5), (0, 6), (0, 7), (1, 4), (1, 5), (2, 3)
(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1)
Prentice Hall Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
30
Name Class Date
Solving a System by Graphing
Problem
What is the solution of the system of inequalities? e 2x 2 y . 1
2x 1 y $ 3Step 1 Solve each inequality for y. 2x 2 y . 1 x 1 y $ 3
2y . 22x 1 1 and y $ 2x 1 3y , 2x 2 1
Step 2 Graph the boundary lines. Use a solid line for $ or #
xO 2 4�4 �2
�2
2
4
�4
y
inequalities. Use a dotted line for . and , inequalities.
Step 3 Shade on the appropriate side of each boundary line.
xO 2 4�4 �2
�2
2
4
�4
y
Th e overlap is the solution to the system.
Exercises
Solve each system of inequalities by graphing.
5. e y # xy $ 3x 2 1
6. e 2x 1 y . 3
2x 2 y , 2 7. e x . 1
y , x 1 1
8. e 2x 1 3y # 9
2x 2 3y . 1 9. e y , 2
13x 2 1
y $ 3x 1 1 10. e 4x 1 2y # 1
4x 1 2y # 21
11. e y # 3
y . 4x 2 3 12. e 2x 1 y , 3
3x 2 y , 2 13. e y $ 22
3x 1 y # 2
3-3 Reteaching (continued) Systems of Inequalities
O
2
�2�2
2x
y
O
2
�2�2
2
x
y
O
2
�2�2
2
x
y
O
2
�2�2
2x
y
O
2
�2�2
2 x
y
O2
�2�2
2
x
y
O
2
�2�2
2x
y
O
2
�2�2
2
x
y
O
2
�2�2
2
x
y
Prentice Hall Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
39
Name Class Date
Problem
What point in the feasible region maximizes P for the objective function P 5 10x 1 15y? What point minimizes P?
Constraints dx 1 y # 16
3x 1 6y # 60
x $ 0
y $ 0
Step 1 Step 2 Step 3
Graph the constraints and shade the feasible region.
Find the coordinates for each vertex of the region.
Evaluate P at each vertex.
VERTEX P 5 10x 1 15y
A (0, 0) P 5 10(0) 1 15(0) 5 0
B (16, 0) P 5 10(16) 1 15(0) 5 160
C (12, 4) P 5 10(12) 1 15(4) 5 180
D (0, 10) P 5 10(0) 1 15(10) 5 150
Th e maximum value of the objective function is 180. It occurs when x 5 12 and y 5 4.
Th e minimum value of the objective function is 0. It occurs when x 5 0 and y 5 0.
Exercises
Graph each system of constraints. Name all vertices. Th en fi nd the values of x and y that maximize or minimize the objective function.
1. d5y 1 4x # 35
5y 1 x $ 20
y # 6
x $ 1
2. dx 1 y $ 2
x $ yx # 4
y $ 0
3. c3x 1 4y $ 12
5x 1 6y # 30
1 # x # 3
P 5 8x 1 2y P 5 x 1 3y P 5 x 2 2y
3-4 ReteachingLinear Programming
xO
y
D
C
BA
8
8
16
vertices: (1, 6), Q1, 195 R ,
Q54, 6R , (5, 3)
max: 46 at (5, 3);
min: 785 at Q1, 19
5 R
vertices: (4, 0), (1, 1), (2, 0), (4, 4)
max: 16 at (4, 4); min: 2 at (2, 0)
vertices: Q1, 94R , Q1, 25
6 R , Q3, 3
4R , Q3, 52R
y
x2
O
4
6
4 62
(5, 3)
(1, 6)(5/4, 6)
(1, 19/5)
y
x2
O
4
6
4 62(2, 0)
(4, 4)
(4, 0)
(1, 1) y
x2
O
4
6
4 62
(1, 9/4)
(1, 3/6)
(3, 5/2)
max: 32 at Q3, 34R ;
min: 2 223 at Q1, 25
6 R
Prentice Hall Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
40
Name Class Date
Problem
Your school band is selling calendars as a fundraiser. Wall calendars cost $48 per case of 24. You sell them at $7 per calendar. Pocket calendars cost $30 per case of 40. You sell them at $3 per calendar. You make a profi t of $120 per case of wall calendars and $90 per case of pocket calendars. If the band can buy no more than 1000 total calendars and spend no more than $1200, how can you maximize your profi t if you sell every calendar? What is the maximum profi t?
Relate Organize the information in a table.
Defi ne Let x 5 number of cases of wall calendars Let y 5 number of cases of pocket calendarsWrite Use the information in the table and the defi nitions of x and y to write the constraints
and the objective function. Simplify the inequalities if necessary. 24x 1 40y # 1000 48x 1 30y # 1200 23x 1 45y # 125 8x 1 5y # 200
c3x 1 5y # 125
8x 1 5y # 200
x $ 0, y $ 0
Objective function: P 5 120x 1 90y
Step 1 Step 2 Step 3Graph the constraints and shade to see the feasible region.
Find the coordinates for each vertex of the region.
Evaluate the objective function using the vertex coordinates.
A(0, 0) P 5 120(0) 1 90(0) 5 0
B(25, 0) P 5 120(25) 1 90(0) 5 3000
C(15, 16) P 5 120(15) 1 90(16) 5 3240
D(0, 25) P 5 120(0) 1 90(25) 5 2250
You can maximize your profi t by selling 15 cases of wall calendars and 16 cases of pocket calendars. Th e maximum profi t is $3240.
Exercises 4. Your band decides to sell the wall calendars for $9 each. a. How many of each type of calendar should you now buy to maximize your
profi t? b. What is the maximum profi t?
4
48
12162024
8 12 16 20 24
y
xO
3-4 Reteaching (continued)
Linear Programming
Number of Cases
Number of Units
Cost
Profit
Wall Calendars
x
24x
48x
120x
Pocket Calendars
y
40y
30y
90y
Total
1000
1200
120x 1 90y
25 cases of wall calendars and no cases of pocket calendars$4200
Prentice Hall Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
49
Name Class Date
Problem
What is the solution of the system?
•x 1 2y 1 3z 5 216
2x 2 2y 1 3z 5 219
2x 1 2y 1 2z 5 219
Use elimination. Th e equations are numbered to make the process easy to follow.
x 1 2y 1 2z 5 216
2x 1 2y 1 2z 5 219
3y 1 3z 5 215
2x 2 2y 1 3z 5 19
2x 1 2y 1 2z 5 29
2x 2 4y 1 3z 5 219
22x 1 4y 1 4z 5 218
3y 1 7z 5 227
3y 1 (23z) 5 215
23y 1 (27z) 5 227
24z) 5 212
z) 5 213
3y 1 3(3) 5 215
3y 5 216
y 5 212
x 1 2 1 3 5 216
x 5 211
Th e solution is (1, 2, 3).
ExercisesSolve each system by elimination. Check your answers.
1. •2x 2 3y 1 2z 5 10
4x 1 2y 2 5z 5 10
4x 2 3y 1 5z 5 8
2. •3x 2 3y 1 2z 5 6
2x 1 3y 1 2z 5 2
3x 1 5y 1 4z 5 4
3. •6x 2 4y 1 5z 5 31
5x 1 2y 1 2z 5 13
5x 1 4y 1 5z 5 2
4. •3x 1 2y 1 3z 5 2
4x 2 2y 1 3z 5 24
2x 1 2y 1 2z 5 8
5. •5x 1 2y 1 3z 5 5
3x 2 3y 2 3z 5 9
3x 1 2y 1 4z 5 6
6. •4x 1 3y 1 2z 5 21
4x 1 3y 1 2z 5 210
2x 2 4y 2 2z 5 26
A
B
C
A
C
Pair the equations to eliminate x .
B
C
D
B
C
E
Multiply equation 3 by 2 to eliminate x . Then add the two equations.
D
E
Equations 4 and 5 form a system.
Multiply equation 5 by 21 and add to equation 4 to eliminate y and solve for z.
A
D Substitute z 5 3 into equation 4 and solve for y .
Substitute the values of y and z into one of the original equations. Solve for x .
3-5 ReteachingSystems With Three Variables
Pair a different set of equations.
(23, 22, 4)
(4, 2, 2) (2, 22, 2) (3, 22, 1)
(21, 3, 2) (2, 24, 3)
Prentice Hall Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
50
Name Class Date
Problem
What is the solution of the system? Use substitution. •24x 1 3y 2 2z 5 19
24x 2 2y 1 3z 5 8
23x 1 2y 1 2z 5 15
Step 1 Choose an equation that can be solved easily for one variable. Choose equation 1 and solve for x.
Step 2 Substitute the expression for x into equations 2 and 3 and simplify.
4(23y 1 2z 1 19) 2 2y 1 3z 5 8 23(23y 1 2z 1 19) 1 2y 1 2z 5 15212y 1 8z 1 76 2 2y 1 3z 5 8 9y 2 6z 2 57 1 2y 1 2z 5 15
D 214y 1 11z 5 268 E 11y 2 4z 5 72
Step 3 Write the two new equations as a system. Solve for y and z. e214y 1 11z 5 268
211y 2 14z 5 272
256y 1 44z 5 2272 Multiply D by 4. 214y 1 11z 5 268 Substitute y 5 8 into D.121y 2 44z 5 2792 Multiply E by 11. 214(8) 1 11z 5 268
65y 5 520 11z 5 44y 5 8 z 5 4
Step 4 Use one of the original equations to solve for x.x 1 3y 2 2z 5 19 Substitute y 5 8 and z 5 4 into A.
x 1 3(8) 2 2(4) 5 19x 5 3
Th e solution of the system is (3, 8, 4).
Exercises
Solve each system by substitution. Check your answers.
7. •24x 1 3y 2 2z 5 7
2x 2 2y 1 3z 5 15
2x 1 2y 2 2z 5 26
8. •22x 2 2y 1 5z 5 25
23x 1 2y 2 7z 5 10
22x 2 3y 1 6z 5 212
9. •2x 2 4y 1 3z 5 15
3x 2 4y 2 2z 5 7
2x 1 2y 1 5z 5 22
10. •22x 1 2y 2 2z 5 21
22x 1 2y 1 2z 5 11
24x 2 3y 1 2z 5 4
11. •2x 2 3y 1 2z 5 7
2x 2 5y 2 4z 5 21
2x 1 4y 2 6z 5 22
12. •23x 2 4y 2 3z 5 213
22x 1 4y 2 5z 5 25
25x 1 2y 1 5z 5 3
A
B
C
A x 1 3y 2 2z 5 19
x 5 23y 1 2z 1 19
D
E
3-5 Reteaching (continued)
Systems With Three Variables
(22, 7, 11)
(1, 2, 3)
(3, 4, 1)
(6, 1, 2)
(5, 1, 2)
(2, 4, 5)
Prentice Hall Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
59
Name Class Date
Problem
How can you represent the system of equations with a matrix? c24x 2 3y 1 5z 5 213
24x 1 3y 5 3
22x 1 4y 1 3z 5 17
Step 1 Write each equation in the same variable order. Line up the like variables. Write in variables that have a coeffi cient of 0.
c24x 2 3y 1 5z 5 213
24x 1 3y 1 0z 5 213
22x 1 4y 1 3z 5 217
Step 2 Write the matrix using the coeffi cients and constants. Remember to enter a 1 for variables with no numeric coeffi cient.
D 4 23 5
1 3 0
22 4 3
4 213
3
17
TExercises
Write a matrix to represent each system.
1. e 2x 1 y 5 23
3y 5 5 2. c3x 2 5y 1 2z 5 9
4x 1 7y 1 z 5 3
2x 2 z 5 12
3. c5x 2 y 1 3z 5 2
3y 1 2z 5 6
4x 1 3y 1 z 5 1
4. c2x 2 z 5 3
5y 1 4z 5 25
2x 1 2y 5 1
5. cz 5 6
x 1 y 5 2
3x 2 2y 2 5z 5 10
6. c2z 2 5x 1 3y 5 4
2y 1 2x 1 4z 5 22
3x 1 z 2 2y 5 25
3-6 ReteachingSolving Systems Using Matrices
c2 10 3
` 235d
D 2 0 210 5 4
21 2 0 4 3
251T
D3 25 24 7 12 0 21
4 93
12T
D0 0 11 1 03 22 25
4 62
10T
D5 21 30 3 24 3 1
4 261T
D25 3 22 21 43 22 1
4 42225T
Prentice Hall Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
60
Name Class Date
Problem
What is the solution of the system? e 2x 1 5y 5 5
2x 1 2y 5 27
Step 1 Write the matrix for the system.
c 2 5
21 2 ` 5
27d
Step 2 Multiply Row 2 by 2. Add to Row 1. Replace Row 1 with the sum. Write the new matrix.
2 5 5)
12(21 2 27)
0 9 29)
c 0 9
21 2 ` 29
27d
Step 3 Divide Row 1 by 9. Write the new matrix.
19(0 9 29) c 0 1
21 2 ` 21
27d
Step 4 Multiply Row 1 by 22. Add to Row 2. Replace Row 2 with the sum. Write the new matrix.
22(0 1 21)
121 2 27)
21 0 25)
c 0 1
21 0 ` 21
25d
Step 5 Multiply Row 2 by 21. Write the new matrix.
21(21 0 25) c0 1
1 0 ` 21
5d
Th is matrix is equivalent to the system e y 5 21
x 5 5. Th e solution is (5, 21).
Exercises
Solve each system of equations using a matrix.
7. e24x 1 3y 5 6
2x 2 y 5 21 8. e 6x 1 y 5 22
2x 1 3y 5 13 9. e 3x 1 2y 5 24
24x 2 3y 5 7
3-6 Reteaching (continued) Solving Systems Using Matrices
(21, 4) (2, 25)(3, 22)