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Elementary Algebra 2e

5.2 Solving Systems of Equations by Substitution

Elementary Algebra 2e5.2 Solving Systems of Equations by Substitution
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Key Terms
    13. Key Concepts
    14. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope-Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solving Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Trinomials of the Form x2+bx+c
    4. 7.3 Factor Trinomials of the Form ax2+bx+c
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Key Terms
    12. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations in Two Variables
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
  13. Index
Be Prepared 5.4

Before you get started, take this readiness quiz.

Simplify −5(3x)−5(3x).
If you missed this problem, review Example 1.136.

Be Prepared 5.5

Simplify 42(n+5)42(n+5).
If you missed this problem, review Example 1.123.

Be Prepared 5.6

Solve for yy. 8y8=322y8y8=322y
If you missed this problem, review Example 2.34.

Be Prepared 5.7

Solve for xx. 3x9y=−33x9y=−3
If you missed this problem, review Example 2.65.

Solving systems of linear equations by graphing is a good way to visualize the types of solutions that may result. However, there are many cases where solving a system by graphing is inconvenient or imprecise. If the graphs extend beyond the small grid with x and y both between −10 and 10, graphing the lines may be cumbersome. And if the solutions to the system are not integers, it can be hard to read their values precisely from a graph.

In this section, we will solve systems of linear equations by the substitution method.

Solve a System of Equations by Substitution

We will use the same system we used first for graphing.

{2x+y=7x2y=6{2x+y=7x2y=6

We will first solve one of the equations for either x or y. We can choose either equation and solve for either variable—but we’ll try to make a choice that will keep the work easy.

Then we substitute that expression into the other equation. The result is an equation with just one variable—and we know how to solve those!

After we find the value of one variable, we will substitute that value into one of the original equations and solve for the other variable. Finally, we check our solution and make sure it makes both equations true.

We’ll fill in all these steps now in Example 5.13.

Example 5.13 How to Solve a System of Equations by Substitution

Solve the system by substitution. {2x+y=7x2y=6{2x+y=7x2y=6

 

 

 

Try It 5.25

Solve the system by substitution. {−2x+y=−11x+3y=9{−2x+y=−11x+3y=9

Try It 5.26

Solve the system by substitution. {x+3y=104x+y=18{x+3y=104x+y=18

How To

Solve a system of equations by substitution.

  1. Step 1. Solve one of the equations for either variable.
  2. Step 2. Substitute the expression from Step 1 into the other equation.
  3. Step 3. Solve the resulting equation.
  4. Step 4. Substitute the solution in Step 3 into one of the original equations to find the other variable.
  5. Step 5. Write the solution as an ordered pair.
  6. Step 6. Check that the ordered pair is a solution to both original equations.

If one of the equations in the system is given in slope–intercept form, Step 1 is already done! We’ll see this in Example 5.14.

Example 5.14

Solve the system by substitution.

{x+y=−1y=x+5{x+y=−1y=x+5

Try It 5.27

Solve the system by substitution. {x+y=6y=3x2{x+y=6y=3x2

Try It 5.28

Solve the system by substitution. {2xy=1y=−3x6{2xy=1y=−3x6

If the equations are given in standard form, we’ll need to start by solving for one of the variables. In this next example, we’ll solve the first equation for y.

Example 5.15

Solve the system by substitution. {3x+y=52x+4y=−10{3x+y=52x+4y=−10

Try It 5.29

Solve the system by substitution. {4x+y=23x+2y=−1{4x+y=23x+2y=−1

Try It 5.30

Solve the system by substitution. {x+y=44xy=2{x+y=44xy=2

In Example 5.15 it was easiest to solve for y in the first equation because it had a coefficient of 1. In Example 5.16 it will be easier to solve for x.

Example 5.16

Solve the system by substitution. {x2y=−23x+2y=34{x2y=−23x+2y=34

Try It 5.31

Solve the system by substitution. {x5y=134x3y=1{x5y=134x3y=1

Try It 5.32

Solve the system by substitution. {x6y=−62x4y=4{x6y=−62x4y=4

When both equations are already solved for the same variable, it is easy to substitute!

Example 5.17

Solve the system by substitution. {y=−2x+5y=12x{y=−2x+5y=12x

Try It 5.33

Solve the system by substitution. {y=3x16y=13x{y=3x16y=13x

Try It 5.34

Solve the system by substitution. {y=x+10y=14x{y=x+10y=14x

Be very careful with the signs in the next example.

Example 5.18

Solve the system by substitution. {4x+2y=46xy=8{4x+2y=46xy=8

Try It 5.35

Solve the system by substitution. {x4y=−4−3x+4y=0{x4y=−4−3x+4y=0

Try It 5.36

Solve the system by substitution. {4xy=02x3y=5{4xy=02x3y=5

In Example 5.19, it will take a little more work to solve one equation for x or y.

Example 5.19

Solve the system by substitution. {4x3y=615y20x=−30{4x3y=615y20x=−30

Try It 5.37

Solve the system by substitution. {2x3y=12−12y+8x=48{2x3y=12−12y+8x=48

Try It 5.38

Solve the system by substitution. {5x+2y=12−4y10x=−24{5x+2y=12−4y10x=−24

Look back at the equations in Example 5.19. Is there any way to recognize that they are the same line?

Let’s see what happens in the next example.

Example 5.20

Solve the system by substitution. {5x2y=−10y=52x{5x2y=−10y=52x

Try It 5.39

Solve the system by substitution. {3x+2y=9y=32x+1{3x+2y=9y=32x+1

Try It 5.40

Solve the system by substitution. {5x3y=2y=53x4{5x3y=2y=53x4

Solve Applications of Systems of Equations by Substitution

We’ll copy here the problem solving strategy we used in the Solving Systems of Equations by Graphing section for solving systems of equations. Now that we know how to solve systems by substitution, that’s what we’ll do in Step 5.

How To

How to use a problem solving strategy for systems of linear equations.

  1. Step 1. Read the problem. Make sure all the words and ideas are understood.
  2. Step 2. Identify what we are looking for.
  3. Step 3. Name what we are looking for. Choose variables to represent those quantities.
  4. Step 4. Translate into a system of equations.
  5. Step 5. Solve the system of equations using good algebra techniques.
  6. Step 6. Check the answer in the problem and make sure it makes sense.
  7. Step 7. Answer the question with a complete sentence.

Some people find setting up word problems with two variables easier than setting them up with just one variable. Choosing the variable names is easier when all you need to do is write down two letters. Think about this in the next example—how would you have done it with just one variable?

Example 5.21

The sum of two numbers is zero. One number is nine less than the other. Find the numbers.

Try It 5.41

The sum of two numbers is 10. One number is 4 less than the other. Find the numbers.

Try It 5.42

The sum of two number is −6. One number is 10 less than the other. Find the numbers.

In the Example 5.22, we’ll use the formula for the perimeter of a rectangle, P = 2L + 2W.

Example 5.22

The perimeter of a rectangle is 88. The length is five more than twice the width. Find the length and the width.

Try It 5.43

The perimeter of a rectangle is 40. The length is 4 more than the width. Find the length and width of the rectangle.

Try It 5.44

The perimeter of a rectangle is 58. The length is 5 more than three times the width. Find the length and width of the rectangle.

For Example 5.23 we need to remember that the sum of the measures of the angles of a triangle is 180 degrees and that a right triangle has one 90 degree angle.

Example 5.23

The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle. Find the measures of both angles.

Try It 5.45

The measure of one of the small angles of a right triangle is 2 more than 3 times the measure of the other small angle. Find the measure of both angles.

Try It 5.46

The measure of one of the small angles of a right triangle is 18 less than twice the measure of the other small angle. Find the measure of both angles.

Example 5.24

Heather has been offered two options for her salary as a trainer at the gym. Option A would pay her $25,000 plus $15 for each training session. Option B would pay her $10,000 + $40 for each training session. How many training sessions would make the salary options equal?

Try It 5.47

Geraldine has been offered positions by two insurance companies. The first company pays a salary of $12,000 plus a commission of $100 for each policy sold. The second pays a salary of $20,000 plus a commission of $50 for each policy sold. How many policies would need to be sold to make the total pay the same?

Try It 5.48

Kenneth currently sells suits for company A at a salary of $22,000 plus a $10 commission for each suit sold. Company B offers him a position with a salary of $28,000 plus a $4 commission for each suit sold. How many suits would Kenneth need to sell for the options to be equal?

Media Access Additional Online Resources

Access these online resources for additional instruction and practice with solving systems of equations by substitution.

Section 5.2 Exercises

Practice Makes Perfect

Solve a System of Equations by Substitution

In the following exercises, solve the systems of equations by substitution.

71.

{2x+y=−43x2y=−6{2x+y=−43x2y=−6

72.

{2x+y=−23xy=7{2x+y=−23xy=7

73.

{x2y=−52x3y=−4{x2y=−52x3y=−4

74.

{x3y=−92x+5y=4{x3y=−92x+5y=4

75.

{5x2y=−6y=3x+3{5x2y=−6y=3x+3

76.

{−2x+2y=6y=−3x+1{−2x+2y=6y=−3x+1

77.

{2x+3y=3y=x+3{2x+3y=3y=x+3

78.

{2x+5y=−14y=−2x+2{2x+5y=−14y=−2x+2

79.

{2x+5y=1y=13x2{2x+5y=1y=13x2

80.

{3x+4y=1y=25x+2{3x+4y=1y=25x+2

81.

{3x2y=6y=23x+2{3x2y=6y=23x+2

82.

{−3x5y=3y=12x5{−3x5y=3y=12x5

83.

{2x+y=10x+y=−5{2x+y=10x+y=−5

84.

{−2x+y=10x+2y=16{−2x+y=10x+2y=16

85.

{3x+y=1−4x+y=15{3x+y=1−4x+y=15

86.

{x+y=02x+3y=−4{x+y=02x+3y=−4

87.

{x+3y=13x+5y=−5{x+3y=13x+5y=−5

88.

{x+2y=−12x+3y=1{x+2y=−12x+3y=1

89.

{2x+y=5x2y=−15{2x+y=5x2y=−15

90.

{4x+y=10x2y=−20{4x+y=10x2y=−20

91.

{y=−2x1y=13x+4{y=−2x1y=13x+4

92.

{y=x6y=32x+4{y=x6y=32x+4

93.

{y=2x8y=35x+6{y=2x8y=35x+6

94.

{y=x1y=x+7{y=x1y=x+7

95.

{4x+2y=88xy=1{4x+2y=88xy=1

96.

{x12y=−12x8y=−6{x12y=−12x8y=−6

97.

{15x+2y=6−5x+2y=−4{15x+2y=6−5x+2y=−4

98.

{2x15y=712x+2y=−4{2x15y=712x+2y=−4

99.

{y=3x6x2y=0{y=3x6x2y=0

100.

{x=2y4x8y=0{x=2y4x8y=0

101.

{2x+16y=8x8y=−4{2x+16y=8x8y=−4

102.

{15x+4y=6−30x8y=−12{15x+4y=6−30x8y=−12

103.

{y=−4x4x+y=1{y=−4x4x+y=1

104.

{y=14xx+4y=8{y=14xx+4y=8

105.

{y=78x+4−7x+8y=6{y=78x+4−7x+8y=6

106.

{y=23x+52x+3y=11{y=23x+52x+3y=11

Solve Applications of Systems of Equations by Substitution

In the following exercises, translate to a system of equations and solve.

107.

The sum of two numbers is 15. One number is 3 less than the other. Find the numbers.

108.

The sum of two numbers is 30. One number is 4 less than the other. Find the numbers.

109.

The sum of two numbers is −26. One number is 12 less than the other. Find the numbers.

110.

The perimeter of a rectangle is 50. The length is 5 more than the width. Find the length and width.

111.

The perimeter of a rectangle is 60. The length is 10 more than the width. Find the length and width.

112.

The perimeter of a rectangle is 58. The length is 5 more than three times the width. Find the length and width.

113.

The perimeter of a rectangle is 84. The length is 10 more than three times the width. Find the length and width.

114.

The measure of one of the small angles of a right triangle is 14 more than 3 times the measure of the other small angle. Find the measure of both angles.

115.

The measure of one of the small angles of a right triangle is 26 more than 3 times the measure of the other small angle. Find the measure of both angles.

116.

The measure of one of the small angles of a right triangle is 15 less than twice the measure of the other small angle. Find the measure of both angles.

117.

The measure of one of the small angles of a right triangle is 45 less than twice the measure of the other small angle. Find the measure of both angles.

118.

Maxim has been offered positions by two car dealers. The first company pays a salary of $10,000 plus a commission of $1,000 for each car sold. The second pays a salary of $20,000 plus a commission of $500 for each car sold. How many cars would need to be sold to make the total pay the same?

119.

Jackie has been offered positions by two cable companies. The first company pays a salary of $ 14,000 plus a commission of $100 for each cable package sold. The second pays a salary of $20,000 plus a commission of $25 for each cable package sold. How many cable packages would need to be sold to make the total pay the same?

120.

Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal?

121.

Mitchell currently sells stoves for company A at a salary of $12,000 plus a $150 commission for each stove he sells. Company B offers him a position with a salary of $24,000 plus a $50 commission for each stove he sells. How many stoves would Mitchell need to sell for the options to be equal?

Everyday Math

122.

When Gloria spent 15 minutes on the elliptical trainer and then did circuit training for 30 minutes, her fitness app says she burned 435 calories. When she spent 30 minutes on the elliptical trainer and 40 minutes circuit training she burned 690 calories. Solve the system {15e+30c=43530e+40c=690{15e+30c=43530e+40c=690 for ee, the number of calories she burns for each minute on the elliptical trainer, and cc, the number of calories she burns for each minute of circuit training.

123.

Stephanie left Riverside, California, driving her motorhome north on Interstate 15 towards Salt Lake City at a speed of 56 miles per hour. Half an hour later, Tina left Riverside in her car on the same route as Stephanie, driving 70 miles per hour. Solve the system {56s=70ts=t+12{56s=70ts=t+12.

  1. for tt to find out how long it will take Tina to catch up to Stephanie.
  2. what is the value of ss, the number of hours Stephanie will have driven before Tina catches up to her?

Writing Exercises

124.

Solve the system of equations
{x+y=10xy=6{x+y=10xy=6

by graphing. by substitution. Which method do you prefer? Why?

125.

Solve the system of equations
{3x+y=12x=y8{3x+y=12x=y8 by substitution and explain all your steps in words.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This figure shows a table with three rows and four columns. The columns are labeled, “I can…,” “Confidently.” “With some help.” and “No - I don’t get it.” The only column with filled in cells below it is labeled “I can…” It reads, “solve a system of equations by substitution.” “solve applications of systems of equations by substitution.”

After reviewing this checklist, what will you do to become confident for all objectives?

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