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Prealgebra 2e

4.7 Solve Equations with Fractions

Prealgebra 2e4.7 Solve Equations with Fractions
  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:

  • Determine whether a fraction is a solution of an equation
  • Solve equations with fractions using the Addition, Subtraction, and Division Properties of Equality
  • Solve equations using the Multiplication Property of Equality
  • Translate sentences to equations and solve
Be Prepared 4.17

Before you get started, take this readiness quiz. If you miss a problem, go back to the section listed and review the material.

Evaluate x+4whenx=−3x+4whenx=−3
 If you missed this problem, review Example 3.23.

Be Prepared 4.18

Solve: 2y−3=9.2y−3=9.
 If you missed this problem, review Example 3.61.

Be Prepared 4.19

Solve: y−3=−9y−3=−9
 If you missed this problem, review Example 4.28.

Determine Whether a Fraction is a Solution of an Equation

As we saw in Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality, a solution of an equation is a value that makes a true statement when substituted for the variable in the equation. In those sections, we found whole number and integer solutions to equations. Now that we have worked with fractions, we are ready to find fraction solutions to equations.

The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, or a fraction.

How To

Determine whether a number is a solution to an equation.

  1. Step 1. Substitute the number for the variable in the equation.
  2. Step 2. Simplify the expressions on both sides of the equation.
  3. Step 3. Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.

Example 4.95

Determine whether each of the following is a solution of x−310=12.x−310=12.

  1. ⓐx=1x=1
  2. ⓑx=45x=45
  3. ⓒx=−45x=−45
Try It 4.189

Determine whether each number is a solution of the given equation.

x−23=16:x−23=16:

  1. ⓐx=1x=1
  2. ⓑx=56x=56
  3. ⓒx=−56x=−56
Try It 4.190

Determine whether each number is a solution of the given equation.

y−14=38:y−14=38:

  1. ⓐy=1y=1
  2. ⓑy=−58y=−58
  3. ⓒy=58y=58

Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

In Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality, we solved equations using the Addition, Subtraction, and Division Properties of Equality. We will use these same properties to solve equations with fractions.

Addition, Subtraction, and Division Properties of Equality

For any numbers a,b,andc,a,b,andc,

ifa=b,thena+c=b+c.ifa=b,thena+c=b+c. Addition Property of Equality
ifa=b,thena−c=b−c.ifa=b,thena−c=b−c. Subtraction Property of Equality
ifa=b,thenac=bc,c≠0.ifa=b,thenac=bc,c≠0. Division Property of Equality
Table 4.3

In other words, when you add or subtract the same quantity from both sides of an equation, or divide both sides by the same quantity, you still have equality.

Example 4.96

Solve: y+916=516.y+916=516.

Try It 4.191

Solve: y+1112=512.y+1112=512.

Try It 4.192

Solve: y+815=415.y+815=415.

We used the Subtraction Property of Equality in Example 4.96. Now we’ll use the Addition Property of Equality.

Example 4.97

Solve: a−59=−89.a−59=−89.

Try It 4.193

Solve: a−35=−85.a−35=−85.

Try It 4.194

Solve: n−37=−97.n−37=−97.

The next example may not seem to have a fraction, but let’s see what happens when we solve it.

Example 4.98

Solve: 10q=44.10q=44.

Try It 4.195

Solve: 12u=−76.12u=−76.

Try It 4.196

Solve: 8m=92.8m=92.

Solve Equations with Fractions Using the Multiplication Property of Equality

Consider the equation x4=3.x4=3. We want to know what number divided by 44 gives 3.3. So to “undo” the division, we will need to multiply by 4.4. 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,a,b, and c,c,

ifa=b,thenac=bc.ifa=b,thenac=bc.

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

Let’s use the Multiplication Property of Equality to solve the equation x7=−9.x7=−9.

Example 4.99

Solve: x7=−9.x7=−9.

Try It 4.197

Solve: f5=−25.f5=−25.

Try It 4.198

Solve: h9=−27.h9=−27.

Example 4.100

Solve: p−8=−40.p−8=−40.

Try It 4.199

Solve: c−7=−35.c−7=−35.

Try It 4.200

Solve: x−11=−12.x−11=−12.

Solve Equations with a Coefficient of −1−1

Look at the equation −y=15.−y=15. Does it look as if yy is already isolated? But there is a negative sign in front of y,y, so it is not isolated.

There are three different ways to isolate the variable in this type of equation. We will show all three ways in Example 4.101.

Example 4.101

Solve: −y=15.−y=15.

Try It 4.201

Solve: −y=48.−y=48.

Try It 4.202

Solve: −c=−23.−c=−23.

Solve Equations with a Fraction Coefficient

When we have an equation with a fraction coefficient we can use the Multiplication Property of Equality to make the coefficient equal to 1.1.

For example, in the equation:

34x=2434x=24

The coefficient of xx is 34.34. To solve for x,x, we need its coefficient to be 1.1. Since the product of a number and its reciprocal is 1,1, our strategy here will be to isolate xx by multiplying by the reciprocal of 34.34. We will do this in Example 4.102.

Example 4.102

Solve: 34x=24.34x=24.

Try It 4.203

Solve: 25n=14.25n=14.

Try It 4.204

Solve: 56y=15.56y=15.

Example 4.103

Solve: −38w=72.−38w=72.

Try It 4.205

Solve: −47a=52.−47a=52.

Try It 4.206

Solve: −79w=84.−79w=84.

Translate Sentences to Equations and Solve

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

Subtraction Property of Equality:
For any real numbers a, b,a, b, and c,c,

if a=b,a=b, then a−c=b−c.a−c=b−c.		 Addition Property of Equality:
For any real numbers a, b,a, b, and c,c,

if a=b,a=b, then a+c=b+c.a+c=b+c.
Division Property of Equality:
For any numbers a, b,a, b, and c,c, where c≠0c≠0

if a=b,a=b, then ac=bcac=bc		 Multiplication Property of Equality:
For any real numbers a, b,a, b, and cc

if a=b,a=b, then ac=bcac=bc

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

In the next few examples, we’ll translate sentences into equations and then solve the equations. It might be helpful to review the translation table in Evaluate, Simplify, and Translate Expressions.

Example 4.104

Translate and solve: nn divided by 66 is −24.−24.

Try It 4.207

Translate and solve: nn divided by 77 is equal to −21.−21.

Try It 4.208

Translate and solve: nn divided by 88 is equal to −56.−56.

Example 4.105

Translate and solve: The quotient of qq and −5−5 is 70.70.

Try It 4.209

Translate and solve: The quotient of qq and −8−8 is 72.72.

Try It 4.210

Translate and solve: The quotient of pp and −9−9 is 81.81.

Example 4.106

Translate and solve: Two-thirds of ff is 18.18.

Try It 4.211

Translate and solve: Two-fifths of ff is 16.16.

Try It 4.212

Translate and solve: Three-fourths of ff is 21.21.

Example 4.107

Translate and solve: The quotient of mm and 5656 is 34.34.

Try It 4.213

Translate and solve. The quotient of nn and 2323 is 512.512.

Try It 4.214

Translate and solve The quotient of cc and 3838 is 49.49.

Example 4.108

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

Try It 4.215

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

Try It 4.216

Translate and solve: The difference of one-and-three-fourths and xx is five-sixths.

Section 4.7 Exercises

Practice Makes Perfect

Determine Whether a Fraction is a Solution of an Equation

In the following exercises, determine whether each number is a solution of the given equation.

498.

x−25=110:x−25=110:

  1. ⓐx=1x=1
  2. ⓑx=12x=12
  3. ⓒx=−12x=−12
499.

y−13=512:y−13=512:

  1. ⓐy=1y=1
  2. ⓑy=34y=34
  3. ⓒy=−34y=−34
500.

h+34=25:h+34=25:

  1. ⓐ h=1h=1
  2. ⓑ h=720h=720
  3. ⓒ h=−720h=−720
501.

k+25=56:k+25=56:

  1. ⓐk=1k=1
  2. ⓑk=1330k=1330
  3. ⓒk=−1330k=−1330

Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

In the following exercises, solve.

502.

y+13=43y+13=43

503.

m+38=78m+38=78

504.

f+910=25f+910=25

505.

h+56=16h+56=16

506.

a−58=−78a−58=−78

507.

c−14=−54c−14=−54

508.

x−(−320)=−1120x−(−320)=−1120

509.

z−(−512)=−712z−(−512)=−712

510.

n−16=34n−16=34

511.

p−310=58p−310=58

512.

s+(−12)=−89s+(−12)=−89

513.

k+(−13)=−45k+(−13)=−45

514.

5j=175j=17

515.

7k=187k=18

516.

−4w=26−4w=26

517.

−9v=33−9v=33

Solve Equations with Fractions Using the Multiplication Property of Equality

In the following exercises, solve.

518.

f4=−20f4=−20

519.

b3=−9b3=−9

520.

y7=−21y7=−21

521.

x8=−32x8=−32

522.

p−5=−40p−5=−40

523.

q−4=−40q−4=−40

524.

r−12=−6r−12=−6

525.

s−15=−3s−15=−3

526.

−x=23−x=23

527.

−y=42−y=42

528.

−h=−512−h=−512

529.

−k=−1720−k=−1720

530.

45n=2045n=20

531.

310p=30310p=30

532.

38q=−4838q=−48

533.

52m=−4052m=−40

534.

−29a=16−29a=16

535.

−37b=9−37b=9

536.

−611u=−24−611u=−24

537.

−512v=−15−512v=−15

Mixed Practice

In the following exercises, solve.

538.

3x=03x=0

539.

8y=08y=0

540.

4f=454f=45

541.

7g=797g=79

542.

p+23=112p+23=112

543.

q+56=112q+56=112

544.

78m=11078m=110

545.

14n=71014n=710

546.

−25=x+34−25=x+34

547.

−23=y+38−23=y+38

548.

1120=−f1120=−f

549.

815=−d815=−d

Translate Sentences to Equations and Solve

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

550.

nn divided by eight is −16.−16.

551.

nn divided by six is −24.−24.

552.

mm divided by −9−9 is −7.−7.

553.

mm divided by −7−7 is −8.−8.

554.

The quotient of ff and −3−3 is −18.−18.

555.

The quotient of ff and −4−4 is −20.−20.

556.

The quotient of gg and twelve is 8.8.

557.

The quotient of gg and nine is 14.14.

558.

Three-fourths of qq is 12.12.

559.

Two-fifths of qq is 20.20.

560.

Seven-tenths of pp is −63.−63.

561.

Four-ninths of pp is −28.−28.

562.

mm divided by 44 equals negative 6.6.

563.

The quotient of hh and 22 is 43.43.

564.

Three-fourths of zz is 15.15.

565.

The quotient of aa and 2323 is 34.34.

566.

The sum of five-sixths and xx is 12.12.

567.

The sum of three-fourths and xx is 18.18.

568.

The difference of yy and one-fourth is −18.−18.

569.

The difference of yy and one-third is −16.−16.

Everyday Math

570.

Shopping Teresa bought a pair of shoes on sale for $48.$48. The sale price was 2323 of the regular price. Find the regular price of the shoes by solving the equation 23p=4823p=48

571.

Playhouse The table in a child’s playhouse is 3535 of an adult-size table. The playhouse table is 1818 inches high. Find the height of an adult-size table by solving the equation 35h=18.35h=18.

Writing Exercises

572.

Example 4.100 describes three methods to solve the equation −y=15.−y=15. Which method do you prefer? Why?

573.

Richard thinks the solution to the equation 34x=2434x=24 is 16.16. Explain why Richard is wrong.

Self Check

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

.

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

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