Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 4.2 The Substitution and Elimination Methods.

Copyright © 2013, 2009, 2005 Pearson Education, Inc.

Section 4.2

The Substitution and Elimination

Methods


Objectives

• The Substitution Method

• The Elimination Method

• Models and Applications


Example

Solve each system of equations. a. b. c.

Solutiona. The first equation is solved for y, so we substitute 3x for y in the second equation.

Substitute x = 7 into y = 3x and it gives y = 21.The solution is (7, 21).

3

28

y x

x y

3 5

3 7

x y

x y

2 3 6

3 6 12

x y

x y

3

28

y x

x y

283x x

4 28x 7x


Example (cont)

b. Solve the first equation for y.

3 5

3 7

x y

x y

Substitute x = 2 into 3x + y = 5.

3 5

3 5

x y

y x

3 7x y

3 ( ) 73 5xx 3 3 5 7x x

6 12x 2x

( 5

1

3 2)

6 5

y

y

y

The solution is (2, 1).


Example (cont)

c. Solve for x in the second equation.

2 3 6

3 6 12

x y

x y

3 6 12

3 6 12

2 4

x y

x y

x y

Substitute y = 2 into x = 2y + 4x = 0

2 3 6x y

42 2 3 6yy 4 8 3 6y y

8 6y 2y

The solution is (0, 2).2y


The Elimination Method

The elimination (or addition) method is a second way to solve linear systems symbolically.


Example

Solve the system of equations.

SolutionAdding the two equations eliminates the y variable.

1

5

x y

x y

1

5

2 0 6

x y

x y

x y

2 6x

3x

Substitute x = 3 into either equation.

3

1

2

1

x y

y

y



Example


SolutionIf we multiply the first equation by 1 and then add, the x-variable will be eliminated.

3 4 10

3 5 26

x y

x y

3 4 10

3 5 26

x y

x y

3 4 10

3 5 26

x y

x y

9 36y

4y

Substitute y = 4 into either equation.

3 4 10

3 4( ) 10

3 16 1

4

0

6

2

3

x y

x

x

x

x



Example


SolutionMultiply the first equation by 2 and the second equation by 3 to eliminate y.

4 3 13

3 2 9

x y

x y

2(4 3 ) 2( 13)

3( 3 2 ) 3(9)

x y

x y

8 6 26

9 6 27

x y

x y

1

1

x

x

4 3 13

4( ) 3 13

4 3 13

3

9

1

3

x y

y

y

y

y

The solution is (1, 3).Substitute x = 1 into either equation.


Example


SolutionMultiply the first equation by 4.

11

44 3 20

x y

x y

11

41

4 4( 1)4

x y

x y

4 4x y

4 4x y 4 3 20x y

4 24y 6y

Substitute y = 6 into either equation.

4 3 20

4 3( 206)

x y

x

4 18 20x 4 2x

1

2x

The solution is (1/2, 6).


Example

Use elimination to solve the following system.

SolutionMultiply the first equation by 3 and then add.

2 3 7

6 9 21

x y

x y

The statement 0 = 0 is always true, which indicates that the system has infinitely many solutions. The graphs of these equations are identical lines, and every point on this line represents a solution.

6 9 21

6 9 21

x y

x y

0 0


Example

Use elimination to solve the following system.

SolutionMultiply the second equation by 2 and then add.

The statement 0 = 32 is always false, which tells us that the system has no solutions. These two lines are parallel and they do not intersect.

4 2 14

2 9

x y

x y

4 2 14

4 2 18

x y

x y

0 32


Example

A cruise boat travels 72 miles downstream in 4 hours and returns upstream in 6 hours. Find the rate of the stream. SolutionStep 1: Identify each variable.

Let x = the speed of the boat Let y = the speed of the stream

Step 2: Write the system of linear equations. The boat travels 72 miles downstream in 4 hours.

72/4 = 18 miles per hour. x + y = 18


Example (cont)

The boat travels 72 miles in 6 hours upstream. 72/6 = 12 miles per hour. x – y = 12Step 3a: Solve the system of linear equations.

Step 3b: Determine the solution to the problem.The rate of the stream is 3 mph.

18

12

x y

x y

2 30x

18

15 18

3

x y

y

y

15x


Example (cont)

Step 4: Check your answer.

15 3 18

15 3 12

72/4 = 18 miles per hour

72/6 = 12 miles per hour

The answer checks.


Example

Suppose that two groups of students go to a basketball game. The first group buys 4 tickets and 2 bags of popcorn for $14, and the second group buys 5 tickets and 5 bags of popcorn for $25. Find the price of a ticket and the price of a bag of popcorn.SolutionStep 1: Identify each variable.

x: cost of a ticket y: cost of a bag of popcorn


Example (cont)

Step 2: Write a system of equations. The first group purchases 4 tickets and 2 bags of popcorn for $14. The second group purchases 5 tickets and 5 bags of popcorn for $25. 4x + 2y = 145x + 5y = 25Step 3: Solve the system of linear equations. Solve the first equation for y.

y = −2x + 7

4x + 2y = 142y = −4x + 14


Example (cont)

Substitute for y in the second equation.

Step 3: Solve the system of linear equations.

4x + 2y = 14

5x + 5y = 25

5x + 5y = 255x + 5(−2x + 7) = 25

5x + (−10x) + 35 = 25−5x = −10

x = 2

Because

y = −2(2) + 7y = 3


Example (cont)

Step 3: Determine the solution to the problem.

The tickets cost $2 each and a bag of popcorn costs $3.

Step 4: Check the solution. The first group purchases 4 at $2 each and 2 bags of popcorn at $3 each which is equal to $14. The second group purchases 5 tickets at $2 each and 5 bags of popcorn for $3 each and this does total $25. The answers check.

4x + 2y = 14

5x + 5y = 25


Example

Walt made an extra $9000 last year from a part-time job. He invested part of the money at 10% and the rest at 6%. He made a total of $780 in interest. How much was invested at 6%? SolutionStep 1 Let x be the amount invested at 10%

Let y be the amount invested at 6%

Step 2 Write the data in a system of equations. 9000

0.10 0.06 780

x y

x y


Example (cont)

Step 3 Solve the system.9000

0.10 0.06 780

x y

x y

10 10 90,000

10 6 78,000

x y

x y

10 10 9000

100 0.10 0.06 100 780

x y

x y

Multiply equation 1 by –10

Multiply equation 2 by 100.

4 12,000 y

3000y

9000

3000 9000

6000

x y

x

x

The amount invested at 6% is $3000. The amount invested at 10% is $6000. The check is left to the student.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 4.2 The Substitution and Elimination Methods.

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