copyright © 2013, 2009, 2005 pearson education, inc. section 4.2 the substitution and elimination...
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Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Section 4.2
The Substitution and Elimination
Methods
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Objectives
• The Substitution Method
• The Elimination Method
• Models and Applications
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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).
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
The Elimination Method
The elimination (or addition) method is a second way to solve linear systems symbolically.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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
The solution is (3, 2).
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Solve the system of equations.
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
The solution is (2, 4).
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Solve the system of equations.
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.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Solve the system of equations.
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).
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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
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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
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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.
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