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Elementary Algebra 2e

2.3 Solve Equations with Variables and Constants on Both Sides

Elementary Algebra 2e2.3 Solve Equations with Variables and Constants on Both Sides
  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 an equation with constants on both sides
  • Solve an equation with variables on both sides
  • Solve an equation with variables and constants on both sides
Be Prepared 2.7

Before you get started, take this readiness quiz.

Simplify: 4y9+9.4y9+9.
If you missed this problem, review Example 1.129.

Solve Equations with Constants on Both Sides

In all the equations we have solved so far, all the variable terms were on only one side of the equation with the constants on the other side. This does not happen all the time—so now we will learn to solve equations in which the variable terms, or constant terms, or both are on both sides of the equation.

Our strategy will involve choosing one side of the equation to be the “variable side”, and the other side of the equation to be the “constant side.” Then, we will use the Subtraction and Addition Properties of Equality to get all the variable terms together on one side of the equation and the constant terms together on the other side.

By doing this, we will transform the equation that began with variables and constants on both sides into the form ax=b.ax=b. We already know how to solve equations of this form by using the Division or Multiplication Properties of Equality.

Example 2.27

Solve: 7x+8=−13.7x+8=−13.

Try It 2.53

Solve: 3x+4=−8.3x+4=−8.

Try It 2.54

Solve: 5a+3=−37.5a+3=−37.

Example 2.28

Solve: 8y9=31.8y9=31.

Try It 2.55

Solve: 5y9=16.5y9=16.

Try It 2.56

Solve: 3m8=19.3m8=19.

Solve Equations with Variables on Both Sides

What if there are variables on both sides of the equation? For equations like this, begin as we did above—choose a “variable” side and a “constant” side, and then use the subtraction and addition properties of equality to collect all variables on one side and all constants on the other side.

Example 2.29

Solve: 9x=8x6.9x=8x6.

Try It 2.57

Solve: 6n=5n10.6n=5n10.

Try It 2.58

Solve: −6c=−7c1.−6c=−7c1.

Example 2.30

Solve: 5y9=8y.5y9=8y.

Try It 2.59

Solve: 3p14=5p.3p14=5p.

Try It 2.60

Solve: 8m+9=5m.8m+9=5m.

Example 2.31

Solve: 12x=x+26.12x=x+26.

Try It 2.61

Solve: 12j=−4j+32.12j=−4j+32.

Try It 2.62

Solve: 8h=−4h+12.8h=−4h+12.

Solve Equations with Variables and Constants on Both Sides

The next example will be the first to have variables and constants on both sides of the equation. It may take several steps to solve this equation, so we need a clear and organized strategy.

Example 2.32 How to Solve Equations with Variables and Constants on Both Sides

Solve: 7x+5=6x+2.7x+5=6x+2.

Try It 2.63

Solve: 12x+8=6x+2.12x+8=6x+2.

Try It 2.64

Solve: 9y+4=7y+12.9y+4=7y+12.

We’ll list the steps below so you can easily refer to them. But we’ll call this the ‘Beginning Strategy’ because we’ll be adding some steps later in this chapter.

How To

Beginning Strategy for Solving Equations with Variables and Constants on Both Sides of the Equation.

  1. Step 1. Choose which side will be the “variable” side—the other side will be the “constant” side.
  2. Step 2. Collect the variable terms to the “variable” side of the equation, using the Addition or Subtraction Property of Equality.
  3. Step 3. Collect all the constants to the other side of the equation, using the Addition or Subtraction Property of Equality.
  4. Step 4. Make the coefficient of the variable equal 1, using the Multiplication or Division Property of Equality.
  5. Step 5. Check the solution by substituting it into the original equation.

In Step 1, a helpful approach is to make the “variable” side the side that has the variable with the larger coefficient. This usually makes the arithmetic easier.

Example 2.33

Solve: 8n4=−2n+6.8n4=−2n+6.

Try It 2.65

Solve: 8q5=−4q+7.8q5=−4q+7.

Try It 2.66

Solve: 7n3=n+3.7n3=n+3.

Example 2.34

Solve: 7a3=13a+7.7a3=13a+7.

Try It 2.67

Solve: 2a2=6a+18.2a2=6a+18.

Try It 2.68

Solve: 4k1=7k+17.4k1=7k+17.

In the last example, we could have made the left side the “variable” side, but it would have led to a negative coefficient on the variable term. (Try it!) While we could work with the negative, there is less chance of errors when working with positives. The strategy outlined above helps avoid the negatives!

To solve an equation with fractions, we just follow the steps of our strategy to get the solution!

Example 2.35

Solve: 54x+6=14x2.54x+6=14x2.

Try It 2.69

Solve: 78x12=18x2.78x12=18x2.

Try It 2.70

Solve: 76y+11=16y+8.76y+11=16y+8.

We will use the same strategy to find the solution for an equation with decimals.

Example 2.36

Solve: 7.8x+4=5.4x8.7.8x+4=5.4x8.

Try It 2.71

Solve: 2.8x+12=−1.4x9.2.8x+12=−1.4x9.

Try It 2.72

Solve: 3.6y+8=1.2y4.3.6y+8=1.2y4.

Section 2.3 Exercises

Practice Makes Perfect

Solve Equations with Constants on Both Sides

In the following exercises, solve the following equations with constants on both sides.

174.

9x3=609x3=60

175.

12x8=6412x8=64

176.

14w+5=11714w+5=117

177.

15y+7=9715y+7=97

178.

2a+8=−282a+8=−28

179.

3m+9=−153m+9=−15

180.

−62=8n6−62=8n6

181.

−77=9b5−77=9b5

182.

35=−13y+935=−13y+9

183.

60=−21x2460=−21x24

184.

−12p9=9−12p9=9

185.

−14q2=16−14q2=16

Solve Equations with Variables on Both Sides

In the following exercises, solve the following equations with variables on both sides.

186.

19z=18z719z=18z7

187.

21k=20k1121k=20k11

188.

9x+36=15x9x+36=15x

189.

8x+27=11x8x+27=11x

190.

c=−3c20c=−3c20

191.

b=−4b15b=−4b15

192.

9q=442q9q=442q

193.

5z=398z5z=398z

194.

6y+12=5y6y+12=5y

195.

4x+34=3x4x+34=3x

196.

−18a8=−22a−18a8=−22a

197.

−11r8=−7r−11r8=−7r

Solve Equations with Variables and Constants on Both Sides

In the following exercises, solve the following equations with variables and constants on both sides.

198.

8x15=7x+38x15=7x+3

199.

6x17=5x+26x17=5x+2

200.

26+13d=14d+1126+13d=14d+11

201.

21+18f=19f+1421+18f=19f+14

202.

2p1=4p332p1=4p33

203.

12q5=9q2012q5=9q20

204.

4a+5=a404a+5=a40

205.

8c+7=−3c378c+7=−3c37

206.

5y30=−5y+305y30=−5y+30

207.

7x17=−8x+137x17=−8x+13

208.

7s+12=5+4s7s+12=5+4s

209.

9p+14=6+4p9p+14=6+4p

210.

2z6=23z2z6=23z

211.

3y4=12y3y4=12y

212.

53c3=23c1653c3=23c16

213.

74m7=34m1374m7=34m13

214.

825q=35q+6825q=35q+6

215.

1115a=45a+41115a=45a+4

216.

43n+9=13n943n+9=13n9

217.

54a+15=34a554a+15=34a5

218.

14y+7=34y314y+7=34y3

219.

35p+2=45p135p+2=45p1

220.

14n+8.25=9n+19.6014n+8.25=9n+19.60

221.

13z+6.45=8z+23.7513z+6.45=8z+23.75

222.

2.4w100=0.8w+282.4w100=0.8w+28

223.

2.7w80=1.2w+102.7w80=1.2w+10

224.

5.6r+13.1=3.5r+57.25.6r+13.1=3.5r+57.2

225.

6.6x18.9=3.4x+54.76.6x18.9=3.4x+54.7

Everyday Math

226.

Concert tickets At a school concert the total value of tickets sold was $1506. Student tickets sold for $6 and adult tickets sold for $9. The number of adult tickets sold was 5 less than 3 times the number of student tickets. Find the number of student tickets sold, s, by solving the equation 6s+27s45=15066s+27s45=1506.

227.

Making a fence Jovani has 150 feet of fencing to make a rectangular garden in his backyard. He wants the length to be 15 feet more than the width. Find the width, w, by solving the equation 150=2w+30+2w150=2w+30+2w.

Writing Exercises

228.

Solve the equation 65y8=15y+765y8=15y+7 explaining all the steps of your solution as in the examples in this section.

229.

Solve the equation 10x+14=−2x+3810x+14=−2x+38 explaining all the steps of your solution as in the examples in this section.

230.

When solving an equation with variables on both sides, why is it usually better to choose the side with the larger coefficient of xx to be the “variable” side?

231.

Is x=−2x=−2 a solution to the equation 52x=−4x+152x=−4x+1 ? 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 four 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 an equation with constants on both sides,” “solve an equation with variables on both sides,” and “solve an equation with variables and constants on both sides. ” 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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