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Prealgebra 2e

8.4 Solve Equations with Fraction or Decimal Coefficients

Prealgebra 2e8.4 Solve Equations with Fraction or Decimal Coefficients
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  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 with fraction coefficients
  • Solve equations with decimal coefficients
Be Prepared 8.10

Before you get started, take this readiness quiz.

Multiply: 8·38.8·38.
If you missed this problem, review Example 4.28

Be Prepared 8.11

Find the LCD of 56and14.56and14.
If you missed this problem, review Example 4.63

Be Prepared 8.12

Multiply: 4.784.78 by 100.100.
If you missed this problem, review Example 5.18

Solve Equations with Fraction Coefficients

Let’s use the General Strategy for Solving Linear Equations introduced earlier to solve the equation 18x+12=14.18x+12=14.

.
To isolate the xx term, subtract 1212 from both sides. .
Simplify the left side. .
Change the constants to equivalent fractions with the LCD. .
Subtract. .
Multiply both sides by the reciprocal of 1818. .
Simplify. .

This method worked fine, but many students don’t feel very confident when they see all those fractions. So we are going to show an alternate method to solve equations with fractions. This alternate method eliminates the fractions.

We will apply the Multiplication Property of Equality and multiply both sides of an equation by the least common denominator of all the fractions in the equation. The result of this operation will be a new equation, equivalent to the first, but with no fractions. This process is called clearing the equation of fractions. Let’s solve the same equation again, but this time use the method that clears the fractions.

Example 8.37

Solve: 18x+12=14.18x+12=14.

Try It 8.73

Solve: 14x+12=58.14x+12=58.

Try It 8.74

Solve: 16y13=16.16y13=16.

Notice in Example 8.37 that once we cleared the equation of fractions, the equation was like those we solved earlier in this chapter. We changed the problem to one we already knew how to solve! We then used the General Strategy for Solving Linear Equations.

How To

Solve equations with fraction coefficients by clearing the fractions.

  1. Step 1. Find the least common denominator of all the fractions in the equation.
  2. Step 2. Multiply both sides of the equation by that LCD. This clears the fractions.
  3. Step 3. Solve using the General Strategy for Solving Linear Equations.

Example 8.38

Solve: 7=12x+34x23x.7=12x+34x23x.

Try It 8.75

Solve: 6=12v+25v34v.6=12v+25v34v.

Try It 8.76

Solve: −1=12u+14u23u.−1=12u+14u23u.

In the next example, we’ll have variables and fractions on both sides of the equation.

Example 8.39

Solve: x+13=16x12.x+13=16x12.

Try It 8.77

Solve: a+34=38a12.a+34=38a12.

Try It 8.78

Solve: c+34=12c14.c+34=12c14.

In Example 8.40, we’ll start by using the Distributive Property. This step will clear the fractions right away!

Example 8.40

Solve: 1=12(4x+2).1=12(4x+2).

Try It 8.79

Solve: −11=12(6p+2).−11=12(6p+2).

Try It 8.80

Solve: 8=13(9q+6).8=13(9q+6).

Many times, there will still be fractions, even after distributing.

Example 8.41

Solve: 12(y5)=14(y1).12(y5)=14(y1).

Try It 8.81

Solve: 15(n+3)=14(n+2).15(n+3)=14(n+2).

Try It 8.82

Solve: 12(m3)=14(m7).12(m3)=14(m7).

Solve Equations with Decimal Coefficients

Some equations have decimals in them. This kind of equation will occur when we solve problems dealing with money and percent. But decimals are really another way to represent fractions. For example, 0.3=3100.3=310 and 0.17=17100.0.17=17100. So, when we have an equation with decimals, we can use the same process we used to clear fractions—multiply both sides of the equation by the least common denominator.

Example 8.42

Solve: 0.8x5=7.0.8x5=7.

Try It 8.83

Solve: 0.6x1=11.0.6x1=11.

Try It 8.84

Solve: 1.2x3=9.1.2x3=9.

Example 8.43

Solve: 0.06x+0.02=0.25x1.5.0.06x+0.02=0.25x1.5.

Try It 8.85

Solve: 0.14h+0.12=0.35h2.4.0.14h+0.12=0.35h2.4.

Try It 8.86

Solve: 0.65k0.1=0.4k0.35.0.65k0.1=0.4k0.35.

The next example uses an equation that is typical of the ones we will see in the money applications in the next chapter. Notice that we will distribute the decimal first before we clear all decimals in the equation.

Example 8.44

Solve: 0.25x+0.05(x+3)=2.85.0.25x+0.05(x+3)=2.85.

Try It 8.87

Solve: 0.25n+0.05(n+5)=2.95.0.25n+0.05(n+5)=2.95.

Try It 8.88

Solve: 0.10d+0.05(d5)=2.15.0.10d+0.05(d5)=2.15.

Section 8.4 Exercises

Practice Makes Perfect

Solve equations with fraction coefficients

In the following exercises, solve the equation by clearing the fractions.

209.

14x12=3414x12=34

210.

34x12=1434x12=14

211.

56y23=3256y23=32

212.

56y13=7656y13=76

213.

12a+38=3412a+38=34

214.

58b+12=3458b+12=34

215.

2=13x12x+23x2=13x12x+23x

216.

2=35x13x+25x2=35x13x+25x

217.

14m45m+12m=−114m45m+12m=−1

218.

56n14n12n=−256n14n12n=−2

219.

x+12=23x12x+12=23x12

220.

x+34=12x54x+34=12x54

221.

13w+54=w1413w+54=w14

222.

32z+13=z2332z+13=z23

223.

12x14=112x+1612x14=112x+16

224.

12a14=16a+11212a14=16a+112

225.

13b+15=25b3513b+15=25b35

226.

13x+25=15x2513x+25=15x25

227.

1=16(12x6)1=16(12x6)

228.

1=15(15x10)1=15(15x10)

229.

14(p7)=13(p+5)14(p7)=13(p+5)

230.

15(q+3)=12(q3)15(q+3)=12(q3)

231.

12(x+4)=3412(x+4)=34

232.

13(x+5)=5613(x+5)=56

Solve Equations with Decimal Coefficients

In the following exercises, solve the equation by clearing the decimals.

233.

0.6y+3=90.6y+3=9

234.

0.4y4=20.4y4=2

235.

3.6j2=5.23.6j2=5.2

236.

2.1k+3=7.22.1k+3=7.2

237.

0.4x+0.6=0.5x1.20.4x+0.6=0.5x1.2

238.

0.7x+0.4=0.6x+2.40.7x+0.4=0.6x+2.4

239.

0.23x+1.47=0.37x1.050.23x+1.47=0.37x1.05

240.

0.48x+1.56=0.58x0.640.48x+1.56=0.58x0.64

241.

0.9x1.25=0.75x+1.750.9x1.25=0.75x+1.75

242.

1.2x0.91=0.8x+2.291.2x0.91=0.8x+2.29

243.

0.05n+0.10(n+8)=2.150.05n+0.10(n+8)=2.15

244.

0.05n+0.10(n+7)=3.550.05n+0.10(n+7)=3.55

245.

0.10d+0.25(d+5)=4.050.10d+0.25(d+5)=4.05

246.

0.10d+0.25(d+7)=5.250.10d+0.25(d+7)=5.25

247.

0.05(q5)+0.25q=3.050.05(q5)+0.25q=3.05

248.

0.05(q8)+0.25q=4.100.05(q8)+0.25q=4.10

Everyday Math

249.

Coins Taylor has $2.00$2.00 in dimes and pennies. The number of pennies is 22 more than the number of dimes. Solve the equation 0.10d+0.01(d+2)=20.10d+0.01(d+2)=2 for d,d, the number of dimes.

250.

Stamps Travis bought $9.45$9.45 worth of 49-cent49-cent stamps and 21-cent21-cent stamps. The number of 21-cent21-cent stamps was 55 less than the number of 49-cent49-cent stamps. Solve the equation 0.49s+0.21(s5)=9.450.49s+0.21(s5)=9.45 for s,s, to find the number of 49-cent49-cent stamps Travis bought.

Writing Exercises

251.

Explain how to find the least common denominator of 38,16,and23.38,16,and23.

252.

If an equation has several fractions, how does multiplying both sides by the LCD make it easier to solve?

253.

If an equation has fractions only on one side, why do you have to multiply both sides of the equation by the LCD?

254.

In the equation 0.35x+2.1=3.85,0.35x+2.1=3.85, what is the LCD? How do you know?

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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