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Elementary Algebra

2.2 Solve Equations using the Division and Multiplication Properties of Equality

Elementary Algebra2.2 Solve Equations using the Division and Multiplication Properties of Equality
  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 Solve 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 Quadratic Trinomials with Leading Coefficient 1
    4. 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1
    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
    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

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 require simplification
  • Translate to an equation and solve
  • Translate and solve applications
Be Prepared 2.2

Before you get started, take this readiness quiz.

  1. Simplify: −7(1−7).−7(1−7).
    If you missed this problem, review Example 1.68.
  2. Evaluate 9x+29x+2 when x=−3x=−3.
    If you missed this problem, review Example 1.57.

Solve Equations Using the Division and Multiplication Properties of Equality

You may have noticed that all of the equations we have solved so far have been of the form x+a=bx+a=b or xa=bxa=b. We were able to isolate the variable by adding or subtracting the constant term on the side of the equation with the variable. Now we will see how to solve equations that have a variable multiplied by a constant and so will require division to isolate the variable.

Let’s look at our puzzle again with the envelopes and counters in Figure 2.5.

This image illustrates a workspace divided into two sides. The content of the left side is equal to the content of the right side. On the left side, there are two envelopes each containing an unknown but equal number of counters. On the right side are six counters.
Figure 2.5 The illustration shows a model of an equation with one variable multiplied by a constant. On the left side of the workspace are two instances of the unknown (envelope), while on the right side of the workspace are six counters.

In the illustration there are two identical envelopes that contain the same number of counters. Remember, the left side of the workspace must equal the right side, but the counters on the left side are “hidden” in the envelopes. So how many counters are in each envelope?

How do we determine the number? We have to separate the counters on the right side into two groups of the same size to correspond with the two envelopes on the left side. The 6 counters divided into 2 equal groups gives 3 counters in each group (since 6÷2=36÷2=3).

What equation models the situation shown in Figure 2.6? There are two envelopes, and each contains xx counters. Together, the two envelopes must contain a total of 6 counters.

This image illustrates a workspace divided into two sides. The content of the left side is equal to the content of the right side. On the left side, there are two envelopes each containing an unknown but equal number of counters. On the right side are six counters. Underneath the image is the equation modeled by the counters: 2 x equals 6.
Figure 2.6 The illustration shows a model of the equation 2x=62x=6.
.
If we divide both sides of the equation by 2, as we did with the envelopes and counters, .
we get: .

We found that each envelope contains 3 counters. Does this check? We know 2·3=62·3=6, so it works! Three counters in each of two envelopes does equal six!

This example leads to the Division Property of Equality.

The Division Property of Equality

For any numbers a, b, and c, and c0c0,

Ifa=b,thenac=bcIfa=b,thenac=bc

When you divide both sides of an equation by any non-zero number, you still have equality.

Manipulative Mathematics

Doing the Manipulative Mathematics activity “Division Property of Equality” will help you develop a better understanding of how to solve equations by using the Division Property of Equality.

The goal in solving an equation is to ‘undo’ the operation on the variable. In the next example, the variable is multiplied by 5, so we will divide both sides by 5 to ‘undo’ the multiplication.

Example 2.13

Solve: 5x=−27.5x=−27.

Try It 2.25

Solve: 3y=−41.3y=−41.

Try It 2.26

Solve: 4z=−55.4z=−55.

Consider the equation x4=3x4=3. We want to know what number divided by 4 gives 3. So to “undo” the division, we will need to multiply by 4. The Multiplication Property of Equality will allow us to do this. This property says that if we start with two equal quantities and multiply both by the same number, the results are equal.

The Multiplication Property of Equality

For any numbers a, b, and c,

Ifa=b,thenac=bcIfa=b,thenac=bc

If you multiply both sides of an equation by the same number, you still have equality.

Example 2.14

Solve: y−7=−14.y−7=−14.

Try It 2.27

Solve: a−7=−42.a−7=−42.

Try It 2.28

Solve: b−6=−24.b−6=−24.

Example 2.15

Solve: n=9.n=9.

Try It 2.29

Solve: k=8.k=8.

Try It 2.30

Solve: g=3.g=3.

Example 2.16

Solve: 34x=12.34x=12.

Try It 2.31

Solve: 25n=14.25n=14.

Try It 2.32

Solve: 56y=15.56y=15.

In the next example, all the variable terms are on the right side of the equation. As always, our goal in solving the equation is to isolate the variable.

Example 2.17

Solve: 815=45x.815=45x.

Try It 2.33

Solve: 925=45z.925=45z.

Try It 2.34

Solve: 56=83r.56=83r.

Solve Equations That Require Simplification

Many equations start out more complicated than the ones we have been working with.

With these more complicated equations the first step is to simplify both sides of the equation as much as possible. This usually involves combining like terms or using the distributive property.

Example 2.18

Solve: 1423=12y4y5y.1423=12y4y5y.

Try It 2.35

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

Try It 2.36

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

Example 2.19

Solve: −4(a3)7=25.−4(a3)7=25.

Try It 2.37

Solve: −4(q2)8=24.−4(q2)8=24.

Try It 2.38

Solve: −6(r2)12=30.−6(r2)12=30.

Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference.

Properties of Equality

Subtraction Property of EqualityAddition Property of Equality For any real numbersa,b,andc,For any real numbersa,b,andc, ifa=b,thenac=bc.ifa=b,thena+c=b+c. Division Property of EqualityMultiplication Property of Equality For any numbersa,b,andc,andc0,For any numbersa,b,andc, ifa=b,thenac=bc.ifa=b, thenac=bc.Subtraction Property of EqualityAddition Property of Equality For any real numbersa,b,andc,For any real numbersa,b,andc, ifa=b,thenac=bc.ifa=b,thena+c=b+c. Division Property of EqualityMultiplication Property of Equality For any numbersa,b,andc,andc0,For any numbersa,b,andc, ifa=b,thenac=bc.ifa=b, thenac=bc.

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

Translate to an Equation and Solve

In the next few examples, we will translate sentences into equations and then solve the equations. You might want to review the translation table in the previous chapter.

Example 2.20

Translate and solve: The number 143 is the product of −11−11 and y.

Try It 2.39

Translate and solve: The number 132 is the product of −12 and y.

Try It 2.40

Translate and solve: The number 117 is the product of −13 and z.

Example 2.21

Translate and solve: nn divided by 8 is −32−32.

Try It 2.41

Translate and solve: nn divided by 7 is equal to −21−21.

Try It 2.42

Translate and solve: nn divided by 8 is equal to −56−56.

Example 2.22

Translate and solve: The quotient of yy and −4−4 is 6868.

Try It 2.43

Translate and solve: The quotient of qq and −8−8 is 72.

Try It 2.44

Translate and solve: The quotient of pp and −9−9 is 81.

Example 2.23

Translate and solve: Three-fourths of pp is 18.

Try It 2.45

Translate and solve: Two-fifths of ff is 16.

Try It 2.46

Translate and solve: Three-fourths of ff is 21.

Example 2.24

Translate and solve: The sum of three-eighths and xx is one-half.

Try It 2.47

Translate and solve: The sum of five-eighths and x is one-fourth.

Try It 2.48

Translate and solve: The sum of three-fourths and x is five-sixths.

Translate and Solve Applications

To solve applications using the Division and Multiplication Properties of Equality, we will follow the same steps we used in the last section. We will restate the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve.

Example 2.25

Denae bought 6 pounds of grapes for $10.74. What was the cost of one pound of grapes?

Try It 2.49

Translate and solve:

Arianna bought a 24-pack of water bottles for $9.36. What was the cost of one water bottle?

Try It 2.50

Translate and solve:

At JB’s Bowling Alley, 6 people can play on one lane for $34.98. What is the cost for each person?

Example 2.26

Andreas bought a used car for $12,000. Because the car was 4-years old, its price was 3434 of the original price, when the car was new. What was the original price of the car?

Try It 2.51

Translate and solve:

The annual property tax on the Mehta’s house is $1,800, calculated as 151,000151,000 of the assessed value of the house. What is the assessed value of the Mehta’s house?

Try It 2.52

Translate and solve:

Stella planted 14 flats of flowers in 2323 of her garden. How many flats of flowers would she need to fill the whole garden?

Section 2.2 Exercises

Practice Makes Perfect

Solve Equations Using the Division and Multiplication Properties of Equality

In the following exercises, solve each equation using the Division and Multiplication Properties of Equality and check the solution.

77.

8x=568x=56

78.

7p=637p=63

79.

−5c=55−5c=55

80.

−9x=−27−9x=−27

81.

−809=15y−809=15y

82.

−731=19y−731=19y

83.

−37p=−541−37p=−541

84.

−19m=−586−19m=−586

85.

0.25z=3.250.25z=3.25

86.

0.75a=11.250.75a=11.25

87.

−13x=0−13x=0

88.

24x=024x=0

89.

x4=35x4=35

90.

z2=54z2=54

91.

−20=q−5−20=q−5

92.

c−3=−12c−3=−12

93.

y9=−16y9=−16

94.

q6=−38q6=−38

95.

m−12=45m−12=45

96.

−24=p−20−24=p−20

97.

y=6y=6

98.

u=15u=15

99.

v=−72v=−72

100.

x=−39x=−39

101.

23y=4823y=48

102.

35r=7535r=75

103.

58w=4058w=40

104.

24=34x24=34x

105.

25=110a25=110a

106.

13q=5613q=56

107.

710x=143710x=143

108.

38y=1438y=14

109.

712=34p712=34p

110.

1118=56q1118=56q

111.

518=109u518=109u

112.

720=74v720=74v

Solve Equations That Require Simplification

In the following exercises, solve each equation requiring simplification.

113.

10016=4p10pp10016=4p10pp

114.

−187=5t9t6t−187=5t9t6t

115.

78n34n=9+278n34n=9+2

116.

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

117.

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

118.

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

119.

−10(q4)57=93−10(q4)57=93

120.

−12(d5)29=43−12(d5)29=43

121.

−10(x+4)19=85−10(x+4)19=85

122.

−15(z+9)11=75−15(z+9)11=75

Mixed Practice

In the following exercises, solve each equation.

123.

910x=90910x=90

124.

512y=60512y=60

125.

y+46=55y+46=55

126.

x+33=41x+33=41

127.

w−2=99w−2=99

128.

s−3=−60s−3=−60

129.

27=6a27=6a

130.

a=7a=7

131.

x=2x=2

132.

z16=−59z16=−59

133.

m41=−14m41=−14

134.

0.04r=52.600.04r=52.60

135.

63.90=0.03p63.90=0.03p

136.

−15x=−120−15x=−120

137.

84=−12z84=−12z

138.

19.36=x0.2x19.36=x0.2x

139.

c0.3c=35.70c0.3c=35.70

140.

y=−9y=−9

141.

x=−8x=−8

Translate to an Equation and Solve

In the following exercises, translate to an equation and then solve.

142.

187 is the product of −17−17 and m.

143.

133 is the product of −19−19 and n.

144.

−184−184 is the product of 23 and p.

145.

−152−152 is the product of 8 and q.

146.

u divided by 7 is equal to −49−49.

147.

r divided by 12 is equal to −48−48.

148.

h divided by −13−13 is equal to −65−65.

149.

j divided by −20−20 is equal to −80−80.

150.

The quotient cc and −19−19 is 38.

151.

The quotient of bb and −6−6 is 18.

152.

The quotient of hh and 26 is −52−52.

153.

The quotient kk and 22 is −66−66.

154.

Five-sixths of y is 15.

155.

Three-tenths of x is 15.

156.

Four-thirds of w is 36.

157.

Five-halves of v is 50.

158.

The sum of nine-tenths and g is two-thirds.

159.

The sum of two-fifths and f is one-half.

160.

The difference of p and one-sixth is two-thirds.

161.

The difference of q and one-eighth is three-fourths.

Translate and Solve Applications

In the following exercises, translate into an equation and solve.

162.

Kindergarten Connie’s kindergarten class has 24 children. She wants them to get into 4 equal groups. How many children will she put in each group?

163.

Balloons Ramona bought 18 balloons for a party. She wants to make 3 equal bunches. How many balloons did she use in each bunch?

164.

Tickets Mollie paid $36.25 for 5 movie tickets. What was the price of each ticket?

165.

Shopping Serena paid $12.96 for a pack of 12 pairs of sport socks. What was the price of pair of sport socks?

166.

Sewing Nancy used 14 yards of fabric to make flags for one-third of the drill team. How much fabric, would Nancy need to make flags for the whole team?

167.

MPG John’s SUV gets 18 miles per gallon (mpg). This is half as many mpg as his wife’s hybrid car. How many miles per gallon does the hybrid car get?

168.

Height Aiden is 27 inches tall. He is 3838 as tall as his father. How tall is his father?

169.

Real estate Bea earned $11,700 commission for selling a house, calculated as 61006100 of the selling price. What was the selling price of the house?

Everyday Math

170.

Commission Every week Perry gets paid $150 plus 12% of his total sales amount. Solve the equation 840=150+0.12(a1250)840=150+0.12(a1250) for a, to find the total amount Perry must sell in order to be paid $840 one week.

171.

Stamps Travis bought $9.45 worth of 49-cent stamps and 21-cent stamps. The number of 21-cent stamps was 5 less than the number of 49-cent stamps. Solve the equation 0.49s+0.21(s5)=9.450.49s+0.21(s5)=9.45 for s, to find the number of 49-cent stamps Travis bought.

Writing Exercises

172.

Frida started to solve the equation −3x=36−3x=36 by adding 3 to both sides. Explain why Frida’s method will not solve the equation.

173.

Emiliano thinks x=40x=40 is the solution to the equation 12x=8012x=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.

This is a table that has five rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can...,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can...” reads “1) solve equations using the Division and Multiplication Properties of equality,” “2) solve equations that require simplification,” “3) translate to an equation and solve,” and “4) translate and solve applications.” The rest of the cells are blank.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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