Table of Contents Solve Systems by Graphing Solve Systems by Substitution Solve Systems by Elimination Choosing your Strategy Writing Systems to Model

Table of Contents

Solve Systems by Graphing

Solve Systems by Substitution

Solve Systems by Elimination

Choosing your Strategy

Writing Systems to Model Situations

Solving Systems of Inequalities

Teacher note:Many of the Responder Questions have boxes over the answers. Have the students take some time to solve the system prior to showing them the multiple choice answers by clicking on the box, so they do not just substitute in the answers.T

each

er

note

Click on topic to go to that section.

Strategy One:Graphing

Return to Table of Contents

Some vocabulary...

The "solution" to a system is an ordered pair that will work in each equation. One way to find the solution is to graph the equations on the same coordinate plane and find the point of intersection.

A "system" is two or more linear equations.

Suppose you are walking to school. Your friend is 5 blocks ahead of you. You can walk two blocks per minute, your friend can walk one block per minute. How many minutes will it take for you to catch up with your friend?

Consider this...

Time (min.)

Friend's distance from your start (blocks)

Your distance from your start(blocks)

0 5 0

1 6 2

2 7 4

3 8 6

4 9 8

5 10 10

First, make a table to represent the problem.

Next, plot the points on a graph.

Time (min.)

Blo

cks

05

20

15

10

1510

5

0

Time (min.)

Friend's distance from your start (blocks)

Your distance from your start(blocks)

0 5 0

1 6 2

2 7 4

3 8 6

4 9 8

5 10 10

The point where they intersect is the solution to the system.

Time (min.)

Blo

cks

05

20

15

10

1510

5

0

(5,10) is the solution. In the context of the

problem this means after 5 minutes, you will

meet your friend at block 10.

Solve the system of equations graphically.

y = 2x -3y = x - 1

Solu

tion


2x + y = 3x - 2y = 4

Solu

tion


3x + y = 11x - 2y = 6

Solu

tion

Solve using graphing

y = 4x+6movey = -3x-1move

Write the equation forthe green dashed line

Write the equation forthe blue solid line

What is this pointof intersection?

(move the hand!)(-1, 2)

( , )-1 2

y = 4x+6y = -3x-1

Now take the ordered pair we just found and substitute it into the equation to prove that it is a solution for both lines.

y = 2x + 3

Solve by Graphing

y = -4x - 3

(-1,1)

y= x - 4y= -3x + 4

Solve by Graphing

(2,-2)

What's the problem here?

y= 2x - 4y= 2x + 4

Parallel lines do not intersect!

Therefore there is no solution.

No ordered pair that will work in BOTH equations

( )click to reveal

click to reveal

2y = -4x + 102 2 y = -2x + 5

2x + y = 5 -2x -2x y = -2x + 5

Solve by Graphing

First - transform the equations into y = mx + b form (slope-intercept form)

Now graph the two transformed lines.

2y = 10 -4x becomesy = -2x + 5

2x + y = 5 becomesy = -2x + 5

What's the problem?

The equations transform to the same line.

So we have infinitely

many solutions.

click to reveal click to reveal

1 Solve the system by graphing.y = -x + 4y = 2x +1

A (3,1)

B (1,3)

C (-1,3)

D no solution

Click for multiple choice answers.

Solu

tion

2 Solve the system by graphing.y = 0.5x - 1y = -0.5x -1

A (0,-1)

B (0,0)

C infinitely many

D no solution

Solu

tion


3 Solve the system by graphing.2x + y = 3x - 2y = 4

A (2,4)

B (0.4, 2.2)

C (2, -1)

D no solution

Solu

tion


4 Solve the system by graphing.y = 3x + 3y = 3x - 3

A (0,0)

B (3,3)

C infinitely many

D no solution

Solu

tion


5 Solve the system by graphing.y = 3x + 44y = 12x + 16

A (3,4)

B (-3,-4)

C infinitely many

D no solution

Solu

tion


6 On the accompanying set of axes, graph and label the following lines:  y=5 x = - 4 y = x+5

Calculate the area, in square units, of the triangle formed by the three points of intersection.

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Solu

tion

Strategy Two:Substitution



y = x + 6.1y = -2x - 1.4

NO

TE

Graphing can be inefficient or approximate.

Another way to solve a system is to use substitution.

Substitution allows you to create a one variable equation.

Substitution Explanation

Solve the system using substitution. Why was it difficult to solve this system by graphing?

y = x + 6.1y = -2x - 1.4

y = -2x - 1.4 -start with one equationx + 6.1 = -2x - 1.4 -substitute x + 6.1 for y in equation+2x -6.1 +2x - 6.1 3x = -7.5 -solve for x  x = -2.5

Substitute -2.5 for x in either equation and solve for y. y = x + 6.1 y = (-2.5) + 6.1 y = 3.6

Since x = -2.5 and y = 3.6, the solution is (-2.5, 3.6)

CHECK: See if (-2.5, 3.6) satisfies the other equation. y = -2x - 1.43.6 = -2(-2.5) - 1.43.6 = 5 - 1.43.6 = 3.6

?

?

+ 3x = 21-3 y

y = -2x +14

Solve the system using substitution.

( )

Solu

tion

NO

TE

= -y - 3x

x = -5y - 39

Solve the system using substitution.

( )

Solu

tion

Examine each system of equations.Which variable would you choose to substitute?Why?

y = 4x - 9.6y = -2x + 9

y = -3x7x - y = 42

y = 4x + 1x = 4y + 1

7 Examine the system of equations. Which variable would you substitute?

2x + y = 52y = 10 - 4x

A x

B y

Solu

tion


2y - 8 = xy + 2x = 4

A x

B y

Solu

tion


x - y = 202x + 3y = 0

A x

B y

Solu

tion

Sometimes you need to rewrite one of the equations so that you can use the substitution method. For example:

The system: Is equivalent to:3x -y = 5 y = 3x -52x + 5y = -8 2x + 5y = -8

Using substitution you now have: 2x + 5(3x-5) = -8  -solve for x2x + 15x - 25 = -8 -distribute the 5  17x - 25 = -8 -combine x's  17x = 17 -at 25 to both sides  x = 1 - divide by 17

Substitute x = 1 into one of the equations.2(1) + 5y = -8 2 + 5y = -8 5y = -10 y = -2

The ordered pair (1,-2) satisfies both equations in the original system. 3x -y = 5    2x + 5y = -83(1) - (-2) = 5 2(1) + 5(-2) = -8 3 + 2 = 5  2 - 10 = -8  -8 = -8

Your class of 22 is going on a trip. There are four drivers and two types of vehicles, vans and cars. The vans seat six people, and the cars seat four people, including drivers. How many vans and cars does the class need for the trip?

Let v = the number of vansand c = the number of cars

Set up the system:

 Drivers: v + c = 4 People: 6v + 4c = 22

Solve the system by substitution.  v + c = 4  -solve the first equation for v.    v = -c + 4  -substitute -c + 4 for v in the 6(-c + 4) + 4c = 22  second equation -6c + 24 + 4c = 22  -solve for c  -2c + 24 = 22  -2c = -2  c = 1

  v + c = 4   v + 1 = 4 -substitute for c in the 1st equation  v = 3 -solve for v

Since c = 1 and v = 3, they should use 1 car and 3 vans.

Check the solution in the equations:  v + c = 4 6v + 4c = 22 3 + 1 = 4  6(3) + 4(1) = 22  4 = 4  18 + 4 = 22  22 = 22

Now solve this system using substitution. What happens? x + y = 6 5x + 5y = 10

x + y = 6 -solve the first equation for x  x = 6 - y5(6 - y) + 5y = 10 -substitute 6 - y for x in 2nd equation 30 - 5y + 5y = 10 -solve for y 30 = 10 -FALSE!

Since 30 = 10 is a false statement, the system has no solution.

Now solve this system using substitution. What happens? x + 4y = -3 2x + 8y = -6

x + 4y = -3 - solve the first equation for x  x = -3 - 4y2(-3 - 4y) + 8y = -6 - sub. -3 - 4y for x in 2nd equation -6 - 8y + 8y = -6 - solve for y -6 = -6   - TRUE! - there are infinitely many solutions

How can you quickly decide the number of solutions a system has?

1 Solution Different slopes

No Solution Same slope; different y-intercept (Parallel Lines)

Infinitely Many Same slope; same y-intercept (Same Line)

10 3x - y = -2 y = 3x + 2

A 1 solution

B no solution

C infinitely many solutions

Solu

tion

11  3x + 3y = 8 y = x

A 1 solution

B no solution


1 3

Solu

tion

12  y = 4x  2x - 0.5y = 0

A 1 solution

B no solution


Solu

tion

13 3x + y = 5 6x + 2y = 1

A 1 solution

B no solution


Solu

tion

14  y = 2x - 7 y = 3x + 8

A 1 solution

B no solution


Solu

tion

15 Solve each system by substitution.y = x - 3y = -x + 5

A (4,9)

B (-4,-9)

C (4,1)

D (1,4)

Solu

tion


16 Solve each system by substitution.y = x - 6y = -4

A (-10,-4)

B (-4,2)

C (2,-4)

D (10,4)


Solu

tion

17 Solve each system by substitution.y + 2x = -14y = 2x + 18

A (1,20)

B (1,18)

C (8,-2)

D (-8,2)


Solu

tion

18 Solve each system by substitution.4x = -5y + 50x = 2y - 7

A (6,6.5)

B (5,6)

C (4,5)

D (6,5)

Solu

tion


19 Solve each system by substitution.y = -3x + 23-y + 4x = 19

A (6,5)

B (-7,5)

C (42,-103)

D (6,-5)

Solu

tion


Strategy Three:Elimination


When both linear equations of a system are in Standard Form, Ax + By = C, you can solve the system using elimination.

You can add or subtract the equations to eliminate a variable.

How do you decide which variable to eliminate?

First, look to see if one variable has the same or opposite coefficients. If so, eliminate that variable.

Second, look for which coefficients have a simple least common multiple. Eliminate that variable.

If the variables have the same coefficient, you can subtract the two equations to eliminate the variable.

If the variables have opposite coefficients, you add the two equations to eliminate the variable.

Sometimes, you need to multiply one, or both, equations by a number in order to create a common coefficient.

5x + y = 44-4x - y = -34

Solve by Elimination - Click on the terms to eliminate and they will disappear, then add the two equations

together.

)(

3x + y = 15-3x -3y = -21

Solve by Elimination - Click on the terms and they will disappear then add the two equations together.

( )

5x + y = 17-2x + y = -4

Solve by Elimination - There are 2 ways to complete this problem. See both examples.

( )-1

5x + y = 17 2x - y = 4

( )- 7x = 21

Mult

iplic

ati

on

by -

1

One method is to recognize that the y-coefficient is the same. You can multiply the second equation by -1. This will create opposite coefficients for the y variable. Then, add the two equations.

Subtr

act

ion

5x + y = 17-2x + y = -4

Solve the system by elimination.

 4x + 3y = 16 2x - 3y = 8

Pull

Pull  4x + 3y = 16 4(4) + 3y = 16

+ 2x - 3y = 8  16 + 3y = 16 6x = 24    3y = 0  x = 4  y = 0 (4, 0)

20

A (5,1)

B (-5,-1)

C (1,5)

D no solution

Solve each system by elimination.x + y = 6x - y = 4

Solu

tion


21 Solve each system by elimination.2x + y = -52x - y = -3

A (-2,1)

B (-1,-2)

C (-2,-1)

D infinitely many

Solu

tion


22 Solve each system by elimination.2x + y = -63x + y = -10

A (4,2)

B (3,5)

C (2,4)

D (-4,2)

Solu

tion


23 Solve each system by elimination.4x - y = 5x - y = -7

A no solution

B (4,11)

C (-4,-11)

D (11,-4)

Solu

tion


24 Solve each system by elimination.3x + 6y = 48-5x + 6y = 32

A (2,-7)

B (7,2)

C (2,7)

D infinitely many

Solu

tion


Sometimes, it is not possible to eliminate a variable by adding or subtracting the equations.

When this is the case, you need to multiply one or both equations by a nonzero number in order to create a common coefficient. Then add or subtract the equations.

Examine each system of equations.Which variable would you choose to eliminate?What do you need to multiply each equation by?

2x + 5y = -1 x + 2y = 0

3x + 8y = 815x - 6y = -39

3x + 6y = 62x - 3y = 4

In order to eliminate the y, you need to multiply first.

 3x + 4y = -10 5x - 2y = 18

Multiply the second equation by 2 so the coefficients are opposites. 2(5x - 2y = 18) 10x - 4y = 36

Now solve by adding the equations together. 3x + 4y = -10 10x - 4y = 36 13x = 26  x = 2

Solve for y, by substituting x = 2 into one of the equations.  3x + 4y = -10 3(2) + 4y = -10  6 + 4y = -10  4y = -16  y = -4

So (2,-4) is the solution.

Check:   3x + 4y = -10   5x - 2y = 183(2) + 4(-4) = -10    5(2) - 2(-4) = 18  6 + -16 = -10  10 + 8 = 18 -10 = -10  18 = 18

+

Now solve the same system by eliminating x. What do you multiply the two equations by?

 3x + 4y = -10 5x - 2y = 18

Multiply the first equation by 5 and the second equation by 3 so the coefficients will be the same 5(3x + 4y = -10) 3(5x - 2y = 18)  15x + 20y = -50   15x - 6y = 54

Now solve by subtracting the equations. 15x + 20y = -50  15x - 6y = 54 26y = -104  y = -4

Solve for x, by substituting y = -4 into one of the equations.  3x + 4y = -10 3x + 4(-4) = -10  3x + -16 = -10  3x = 6    x = 2

So (2,-4) is the solution. Check:   3x + 4y = -10   5x - 2y = 183(2) + 4(-4) = -10    5(2) - 2(-4) = 18  6 + -16 = -10  10 + 8 = 18 -10 = -10  18 = 18

-

25 Which variable can you eliminate with the least amount of work?

A x

B y

9x + 6y = 15-4x + y = 3

Solu

tion


A x

B y

3x - 7y = -2-6x + 15y = 9

Solu

tion


A x

B y

x - 3y = -72x + 6y = 34

Solu

tion

28 What will you multiply the first equation by in order to solve this system using elimination?

2x + 5y = 203x - 10y = 37

Now solve it.... (11, ) 25

-

click for answer

(-5,-2)

3x + 2y = -19x - 12y = 19

Now solve it....


click for answer

x + 3y = 43x + 4y = 2

Now solve it.... (-2,2)


click for answer

Choose Your Strategy


Altogether 292 tickets were sold for a basketball game. An adult ticket costs $3. A student ticket costs $1. Ticket sales were $470.

Let a = adults s = students

Set up the system:

 number of tickets sold: a + s = 292 money collected:    3a + s = 470

First eliminate one variable.  a + s = 292 - in both equations s has the same 3a + s = 470 coefficient so you subtract the 2 -2a+ 0 = -178  equations in order to eliminate it. a = 89 -solve for a

Then, find the value of the eliminated variable. a + s = 29289 + s = 292 -substitute 89 for a in 1st equation s = 203 -solve for s

There were 89 adult tickets and 203 student tickets sold.

(89, 203)

Check: a + s = 292  3a + s = 47089 + 203 = 292 3(89) + 203 = 470 292 = 292   267 + 203 = 470  470 = 470

 

-( )

31 A piece of glass with an initial temperature of 99 F is cooled at a rate of 3.5 F/min. At the same time, a piece of copper with an initial temperature of 0 F is heated at a rate of 2.5 F/min. Let m = the number of minutes and t = the temperature in F. Which system models the given information?

A

Solu

tion

B C

t = 99 + 3.5mt = 0 + 2.5m

t = 99 - 3.5mt = 0 + 2.5m

t = 99 + 3.5mt = 0 - 2.5m

32 Which method would you use to solve the system?

A graphing

B substitution

C elimination

t = 99 - 3.5mt = 0 + 2.5m

Now solve it...m = 16.5 t = 41.25

This means that in 16.5 minutes, the temperatures will both be 41.25℃.

click for answer

click for equations

33 What method would you choose to solve the system?

A graphing

B substitution

C elimination

4s - 3t = 8t = -2s -1

Solu

tion

D (-2, )

34 Now solve the system!

A ( , -2) 4s - 3t = 8t = -2s -1

1 2

B ( , 2)

1 2

C (2 , -2)

1 2

Solu

tion



A graphing

B substitution

C elimination

y = 3x - 1y = 4x

Solu

tion

36 Now solve it!

A (1, 4)

B (-4, -1)

C (-1, 4)

y = 3x - 1y = 4x

D (-1, -4)

Solu

tion



A graphing

B substitution

C elimination

3m - 4n = 13m - 2n = -1

Solu

tion

38 Now solve it!

A (-2, -1)

B (-1, -1)

C (-1, 1)

3m - 4n = 13m - 2n = -1

D (1, 1)

Solu

tion



A graphing

B substitution

C elimination

y = -2xy = -0.5x + 3

Solu

tion

40 Now solve it!

A (-6, 12)

B (2, -4)

y = -2xy = -0.5x + 3

C (-2, 4)

D (1, -2)

Solu

tion



A graphing

B substitution

C elimination

2x - y = 4x + 3y = 16

Solu

tion

42 Now solve it!

A (6, 5)

B (-4, 7)

C (-4, 4)

2x - y = 4x + 3y = 16

D (4, 4)


Solu

tion




A graphing

B substitution

C elimination

u = 4v3u - 3v = 7

Solu

tion

44 Now solve it!

A ( , )

B ( , )

C (28, 7)

u = 4v3u - 3v = 7

D (7, ) 7 4

28 9

28 9

7 9

7 9

Solu

tion


45 Choose a strategy and then answer the question.What is the value of the y-coordinate of the solution to the system of equations x − 2y = 1 and x + 4y = 7?

A 1

B -1

C 3

D 4


Solu

tion

ModelingSituations


A group of 148 people is spending five days at a summer camp. The cook ordered 12 pounds of food for each adult and 9 pounds of food for each child. A total of 1,410 pounds of food was ordered.

Part A: Write an equation or a system of equations that describes the above situation and define your variables.


Pull

Pull

Part B: Using your work from part A, find:

(1) the total number of adults in the group

(2) the total number of children in the group

Pull

Pull

Tanisha and Rachel had lunch at the mall. Tanisha ordered three slices of pizza and two colas. Rachel ordered two slices of pizza and three colas. Tanisha’s bill was $6.00, and Rachel’s bill was $5.25. What was the price of one slice of pizza? What was the price of one cola?

Pull

Pull


Sharu has $2.35 in nickels and dimes. If he has a total of thirty-two coins, how many of each coin does he have?

Pull

Pull


Ben had twice as many nickels as dimes. Altogether, Ben had $4.20.How many nickels and how many dimes did Ben have?


Pull

Pull

46 Your class receives $1105 for selling 205 packages of greeting cards and gift wrap. A pack of cards costs $4 and a pack of gift wrap costs $9

Set up a system and solve. How many packages of cards were sold?

You will answer how many packages of gift wrap in the

next question.

Solu

tion

47 Your class receives $1105 for selling 205 packages of greeting cards and gift wrap. A pack of cards costs $4 and a pack of gift wrap costs $9

Set up a system and solve. How many packages of gift wrap were sold? S

olu

tion

48 The sum of two numbers is 47, and their difference is 15. What is the larger number?

A 16

B 31

C 32

D 36


Solu

tion

49 Ramon rented a sprayer and a generator. On his first job, he used each piece of equipment for 6 hours at a total cost of $90. On his second job, he used the sprayer for 4 hours and the generator for 8 hours at a total cost of $100. What was the hourly cost for the sprayer?


Solu

tion

50 What is true of the graphs of the two lines 3y - 8 = -5x and 3x = 2y -18?

A no intersection

B intersect at (2,-6)

C intersect at (-2,6)

D are identical

Solu

tion

51 You have 15 coins in your pocket that are either quarters or nickels. They total $2.75. Set up a system to solve. Which method will you use? (Solving it comes later...)

A graphing

B substitution

C elimination

Solu

tion

52 You have 15 coins in your pocket that are either quarters or nickels. They total $2.75. How many quarters do you have?

Solu

tion

53 You have 15 coins in your pocket that are either quarters or nickels. They total $2.75. How many nickels do you have?

Solu

tion

54 Julia went to the movies and bought one jumbo popcorn and two chocolate chip cookies for $5.00. Marvin went to the same movie and bought one jumbo popcorn and four chocolate chip cookies for $6.00. How much does one chocolate chip cookie cost?

A $0.50

B $0.75

C $1.00

D $2.00


Solu

tion

55 Mary and Amy had a total of 20 yards of material from which to make costumes. Mary used three times more material to make her costume than Amy used, and 2 yards of material was not used. How many yards of material did Amy use for her costume?


Solu

tion

56 The tickets for a dance recital cost $5.00 for adults and $2.00 for children. If the total number of tickets sold was 295 and the total amount collected was $1220, how many adult tickets were sold?


Solu

tion

Solving Systems of Inequalities


Two or more linear inequalities form a system of inequalities.

A solution to the system is an ordered pair that is a solution of each inequality in the system.

Since inequalities have more than one solution, the solutions are best shown in a graph.

Graphing a System of Linear Inequalities

1. Graph the boundary lines of each inequality. (Remember use a dashed line for < and >   and a solid line for < and >)

2. Shade the half-plane for each inequality.

3. The intersection of the half-planes is the solution.

Solve the system of inequalities.

x + 2y < 6-x + y < 0

Pull Pull


2x + y > -4 x - 2y < 4

Pull Pull


4x + 2y < 84x + 2y > -8

Pull Pull

Pull

Pull

Graph the following system of inequalities on the set of axes shown below and label the solution set S.

y > −x + 2

y ≤ 2 x + 5

3

A company manufactures bicycles and skateboards. The company’s daily production of bicycles cannot exceed 10, and its daily production of skateboards must be less than or equal to 12. The combined number of bicycles and skateboards cannot be more than 16. If x is the number of bicycles and y is the number of skateboards, graph on the accompanying set of axes the region that contains the number of bicycles and skateboards the company can manufacture daily.

Pull

Write a system of inequalities from the graph.

Pull

Pull

57 Solve the system of linear inequalities.

A B C

y > -2x + 1y < x + 2

Solu

tion


A B C

x > 2y < 5

Solu

tion


A B C

-2x - 2y < 4y - 2x > 1

Solu

tion


A B C

-5x + y > -24x + y < 1

Solu

tion


A B C

3x + 2y < 122x - 2y < 20 S

olu

tion

62

A (0,4)

B (2,4)

C (-4,1)

D (4,-1)

Which point is in the solution set of the system of inequalities shown in the accompanying graph?


Solu

tion

63 Which ordered pair is in the solution set of the system of inequalities shown in the accompanying graph?

A (0, 0)

B (0, 1)

C (1, 5)

D (3, 2)


Solu

tion

64 Which ordered pair is in the solution set of the following system of linear inequalities?

A (0,3)

B (2,0)

C (−1,0)

D (−1,−4)

y < 2x + 2y ≥ −x − 1


Solu

tion

65 Mr. Braun has $75.00 to spend on pizzas and soda for a picnic. Pizzas cost $9.00 each and the drinks cost $0.75 each. Five times as many drinks as pizzas are needed. What is the maximum number of pizzas that Mr. Braun can buy?


Solu

tion

Documents

Table of Contents Solve Systems by Graphing Solve Systems by Substitution Solve Systems by Elimination Choosing your Strategy Writing Systems to Model