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Prealgebra 2e

8.2 Solve Equations Using the Division and Multiplication Properties of Equality

Prealgebra 2e8.2 Solve Equations Using the Division and Multiplication Properties of Equality
  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. 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
    11. Chapter 11
  17. Index

Learning Objectives

By the end of this section, you will be able to:
  • Solve equations using the Division and Multiplication Properties of Equality
  • Solve equations that need to be simplified
Be Prepared 8.4

Before you get started, take this readiness quiz.

Simplify: −7(1−7).−7(1−7).
If you missed this problem, review Example 4.28.

Be Prepared 8.5

What is the reciprocal of 38?38?
If you missed this problem, review Example 4.29.

Be Prepared 8.6

Evaluate 9x+29x+2 when x=−3.x=−3.
If you missed this problem, review Example 3.56.

Solve Equations Using the Division and Multiplication Properties of Equality

We introduced the Multiplication and Division Properties of Equality in Solve Equations Using Integers; The Division Property of Equality and Solve Equations with Fractions. We modeled how these properties worked using envelopes and counters and then applied them to solving equations (See Solve Equations Using Integers; The Division Property of Equality). We restate them again here as we prepare to use these properties again.

Division and Multiplication Properties of Equality

Division Property of Equality: For all real numbers a,b,c,a,b,c, and c0,c0, if a=b,a=b, then ac=bc.ac=bc.

Multiplication Property of Equality: For all real numbers a,b,c,a,b,c, if a=b,a=b, then ac=bc.ac=bc.

When you divide or multiply both sides of an equation by the same quantity, you still have equality.

Let’s review how these properties of equality can be applied in order to solve equations. Remember, the goal is to ‘undo’ the operation on the variable. In the example below the variable is multiplied by 4,4, so we will divide both sides by 44 to ‘undo’ the multiplication.

Example 8.13

Solve: 4x=−28.4x=−28.

Try It 8.25

Solve: 3y=−48.3y=−48.

Try It 8.26

Solve: 4z=−52.4z=−52.

In the previous example, to ‘undo’ multiplication, we divided. How do you think we ‘undo’ division?

Example 8.14

Solve: a−7=−42.a−7=−42.

Try It 8.27

Solve: b−6=−24.b−6=−24.

Try It 8.28

Solve: c−8=−16.c−8=−16.

Example 8.15

Solve: r=2.r=2.

Try It 8.29

Solve: k=8.k=8.

Try It 8.30

Solve: g=3.g=3.

Example 8.16

Solve: 23x=18.23x=18.

Try It 8.31

Solve: 25n=14.25n=14.

Try It 8.32

Solve: 56y=15.56y=15.

Solve Equations That Need to be Simplified

Many equations start out more complicated than the ones we’ve just solved. First, we need to simplify both sides of the equation as much as possible

Example 8.17

Solve: 8x+9x5x=−3+15.8x+9x5x=−3+15.

Try It 8.33

Solve: 7x+6x4x=−8+26.7x+6x4x=−8+26.

Try It 8.34

Solve: 11n3n6n=717.11n3n6n=717.

Example 8.18

Solve: 1120=17y8y6y.1120=17y8y6y.

Try It 8.35

Solve: 1827=15c9c3c.1827=15c9c3c.

Try It 8.36

Solve: 1822=12xx4x.1822=12xx4x.

Example 8.19

Solve: −3(n2)6=21.−3(n2)6=21.

Try It 8.37

Solve: −4(n2)8=24.−4(n2)8=24.

Try It 8.38

Solve: −6(n2)12=30.−6(n2)12=30.

Section 8.2 Exercises

Practice Makes Perfect

Solve Equations Using the Division and Multiplication Properties of Equality

In the following exercises, solve each equation for the variable using the Division Property of Equality and check the solution.

68.

8x=328x=32

69.

7p=637p=63

70.

−5c=55−5c=55

71.

−9x=−27−9x=−27

72.

−90=6y−90=6y

73.

−72=12y−72=12y

74.

−16p=−64−16p=−64

75.

−8m=−56−8m=−56

76.

0.25z=3.250.25z=3.25

77.

0.75a=11.250.75a=11.25

78.

−3x=0−3x=0

79.

4x=04x=0

In the following exercises, solve each equation for the variable using the Multiplication Property of Equality and check the solution.

80.

x4=15x4=15

81.

z2=14z2=14

82.

−20=q−5−20=q−5

83.

c−3=−12c−3=−12

84.

y9=−6y9=−6

85.

q6=−8q6=−8

86.

m−12=5m−12=5

87.

−4=p−20−4=p−20

88.

23y=1823y=18

89.

35r=1535r=15

90.

58w=4058w=40

91.

24=34x24=34x

92.

25=110a25=110a

93.

13q=5613q=56

Solve Equations That Need to be Simplified

In the following exercises, solve the equation.

94.

8a+3a6a=−17+278a+3a6a=−17+27

95.

6y3y+12y=−43+286y3y+12y=−43+28

96.

−9x9x+2x=502−9x9x+2x=502

97.

−5m+7m8m=−6+36−5m+7m8m=−6+36

98.

10016=4p10pp10016=4p10pp

99.

−187=5t9t6t−187=5t9t6t

100.

78n34n=9+278n34n=9+2

101.

512q+12q=253512q+12q=253

102.

0.25d+0.10d=60.750.25d+0.10d=60.75

103.

0.05p0.01p=2+0.240.05p0.01p=2+0.24

Everyday Math

104.

Balloons Ramona bought 1818 balloons for a party. She wants to make 33 equal bunches. Find the number of balloons in each bunch, b,b, by solving the equation 3b=18.3b=18.

105.

Teaching Connie’s kindergarten class has 2424 children. She wants them to get into 44 equal groups. Find the number of children in each group, g,g, by solving the equation 4g=24.4g=24.

106.

Ticket price Daria paid $36.25$36.25 for 55 children’s tickets at the ice skating rink. Find the price of each ticket, p,p, by solving the equation 5p=36.25.5p=36.25.

107.

Unit price Nishant paid $12.96$12.96 for a pack of 1212 juice bottles. Find the price of each bottle, b,b, by solving the equation 12b=12.96.12b=12.96.

108.

Fuel economy Tania’s SUV gets half as many miles per gallon (mpg) as her husband’s hybrid car. The SUV gets 18 mpg.18 mpg. Find the miles per gallons, m,m, of the hybrid car, by solving the equation 12m=18.12m=18.

109.

Fabric The drill team used 1414 yards of fabric to make flags for one-third of the members. Find how much fabric, f,f, they would need to make flags for the whole team by solving the equation 13f=14.13f=14.

Writing Exercises

110.

Frida started to solve the equation −3x=36−3x=36 by adding 33 to both sides. Explain why Frida’s method will result in the correct solution.

111.

Emiliano thinks x=40x=40 is the solution to the equation 12x=80.12x=80. Explain why he is wrong.

Self Check

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

.

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

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