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Elementary Algebra 2e

2.5 Solve Equations with Fractions or Decimals

Elementary Algebra 2e2.5 Solve Equations with Fractions or Decimals
  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

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 2.13

Before you get started, take this readiness quiz.

Multiply: 8·38.8·38.
If you missed this problem, review Example 1.69.

Be Prepared 2.14

Find the LCD of 5656 and 1414.
If you missed this problem, review Example 1.82.

Be Prepared 2.15

Multiply 4.78 by 100.
If you missed this problem, review Example 1.98.

Solve Equations with Fraction Coefficients

Let’s use the general strategy for solving linear equations introduced earlier to solve the equation, 18x+12=1418x+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 do not 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 without fractions. This process is called “clearing” the equation of fractions.

Let’s solve a similar equation, but this time use the method that eliminates the fractions.

Example 2.48 How to Solve Equations with Fraction Coefficients

Solve: 16y13=5616y13=56.

Try It 2.95

Solve: 14x+12=5814x+12=58.

Try It 2.96

Solve: 18x+12=1418x+12=14.

Notice in Example 2.48, 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

Strategy to solve equations with fraction coefficients.

  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 2.49

Solve: 6=12v+25v34v6=12v+25v34v.

Try It 2.97

Solve: 7=12x+34x23x7=12x+34x23x.

Try It 2.98

Solve: −1=12u+14u23u−1=12u+14u23u.

In the next example, we again have variables on both sides of the equation.

Example 2.50

Solve: a+34=38a12a+34=38a12.

Try It 2.99

Solve: x+13=16x12x+13=16x12.

Try It 2.100

Solve: c+34=12c14c+34=12c14.

In the next example, we start by using the Distributive Property. This step clears the fractions right away.

Example 2.51

Solve: −5=14(8x+4)−5=14(8x+4).

Try It 2.101

Solve: −11=12(6p+2)−11=12(6p+2).

Try It 2.102

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

In the next example, even after distributing, we still have fractions to clear.

Example 2.52

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

Try It 2.103

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

Try It 2.104

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

Example 2.53

Solve: 5x34=x25x34=x2.

Try It 2.105

Solve: 4y73=y64y73=y6.

Try It 2.106

Solve: −2z54=z8−2z54=z8.

Example 2.54

Solve: a6+2=a4+3a6+2=a4+3.

Try It 2.107

Solve: b10+2=b4+5b10+2=b4+5.

Try It 2.108

Solve: c6+3=c3+4c6+3=c3+4.

Example 2.55

Solve: 4q+32+6=3q+544q+32+6=3q+54.

Try It 2.109

Solve: 3r+56+1=4r+333r+56+1=4r+33.

Try It 2.110

Solve: 2s+32+1=3s+242s+32+1=3s+24.

Solve Equations with Decimal Coefficients

Some equations have decimals in them. This kind of equation will occur when we solve problems dealing with money or percentages. But decimals can also be expressed as fractions. For example, 0.3=3100.3=310 and 0.17=171000.17=17100. So, with an equation with decimals, we can use the same method we used to clear fractions—multiply both sides of the equation by the least common denominator.

Example 2.56

Solve: 0.06x+0.02=0.25x1.50.06x+0.02=0.25x1.5.

Try It 2.111

Solve: 0.14h+0.12=0.35h2.40.14h+0.12=0.35h2.4.

Try It 2.112

Solve: 0.65k0.1=0.4k0.350.65k0.1=0.4k0.35.

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

Example 2.57

Solve: 0.25x+0.05(x+3)=2.850.25x+0.05(x+3)=2.85.

Try It 2.113

Solve: 0.25n+0.05(n+5)=2.950.25n+0.05(n+5)=2.95.

Try It 2.114

Solve: 0.10d+0.05(d5)=2.150.10d+0.05(d5)=2.15.

Section 2.5 Exercises

Practice Makes Perfect

Solve Equations with Fraction Coefficients

In the following exercises, solve each equation with fraction coefficients.

318.

14x12=3414x12=34

319.

34x12=1434x12=14

320.

56y23=3256y23=32

321.

56y13=7656y13=76

322.

12a+38=3412a+38=34

323.

58b+12=3458b+12=34

324.

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

325.

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

326.

14m45m+12m=−114m45m+12m=−1

327.

56n14n12n=−256n14n12n=−2

328.

x+12=23x12x+12=23x12

329.

x+34=12x54x+34=12x54

330.

13w+54=w1413w+54=w14

331.

32z+13=z2332z+13=z23

332.

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

333.

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

334.

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

335.

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

336.

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

337.

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

338.

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

339.

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

340.

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

341.

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

342.

5q85=2q105q85=2q10

343.

4m+26=m34m+26=m3

344.

4n+84=n34n+84=n3

345.

3p+63=p23p+63=p2

346.

u34=u23u34=u23

347.

v10+1=v42v10+1=v42

348.

c15+1=c101c15+1=c101

349.

d6+3=d8+2d6+3=d8+2

350.

3x+42+1=5x+1083x+42+1=5x+108

351.

10y23+3=10y+1910y23+3=10y+19

352.

7u141=4u+857u141=4u+85

353.

3v62+5=11v453v62+5=11v45

Solve Equations with Decimal Coefficients

In the following exercises, solve each equation with decimal coefficients.

354.

0.6y+3=90.6y+3=9

355.

0.4y4=20.4y4=2

356.

3.6j2=5.23.6j2=5.2

357.

2.1k+3=7.22.1k+3=7.2

358.

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

359.

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

360.

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

361.

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

362.

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

363.

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

364.

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

365.

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

366.

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

367.

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

368.

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

369.

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

Everyday Math

370.

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

371.

Stamps Paula bought $22.82 worth of 49-cent stamps and 21-cent stamps. The number of 21-cent stamps was 8 less than the number of 49-cent stamps. Solve the equation 0.49s+0.21(s8)=22.820.49s+0.21(s8)=22.82 for s, to find the number of 49-cent stamps Paula bought.

Writing Exercises

372.

Explain how you find the least common denominator of 3838, 1616, and 2323.

373.

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

374.

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

375.

In the equation 0.35x+2.1=3.850.35x+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.

This is a table that has three 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: “solve equations with fraction coefficients,” and “solve equations with decimal coefficients.” The rest of the cells are blank.

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

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