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Elementary Algebra

2.6 Solve a Formula for a Specific Variable

Elementary Algebra2.6 Solve a Formula for a Specific Variable
  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:

  • Use the Distance, Rate, and Time formula
  • Solve a formula for a specific variable
Be Prepared 2.6

Before you get started, take this readiness quiz.

  1. Solve: 15t=120.15t=120.
    If you missed this problem, review Example 2.13.
  2. Solve: 6x+24=96.6x+24=96.
    If you missed this problem, review Example 2.27.

Use the Distance, Rate, and Time Formula

One formula you will use often in algebra and in everyday life is the formula for distance traveled by an object moving at a constant rate. Rate is an equivalent word for “speed.” The basic idea of rate may already familiar to you. Do you know what distance you travel if you drive at a steady rate of 60 miles per hour for 2 hours? (This might happen if you use your car’s cruise control while driving on the highway.) If you said 120 miles, you already know how to use this formula!

Distance, Rate, and Time

For an object moving at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula:

d=rtwhered=distancer=ratet=timed=rtwhered=distancer=ratet=time

We will use the Strategy for Solving Applications that we used earlier in this chapter. When our problem requires a formula, we change Step 4. In place of writing a sentence, we write the appropriate formula. We write the revised steps here for reference.

How To

Solve an application (with a formula).

  1. Step 1. Read the problem. Make sure all the words and ideas are understood.
  2. Step 2. Identify what we are looking for.
  3. Step 3. Name what we are looking for. Choose a variable to represent that quantity.
  4. Step 4. Translate into an equation. Write the appropriate formula for the situation. Substitute in the given information.
  5. Step 5. Solve the equation using good algebra techniques.
  6. Step 6. Check the answer in the problem and make sure it makes sense.
  7. Step 7. Answer the question with a complete sentence.

You may want to create a mini-chart to summarize the information in the problem. See the chart in this first example.

Example 2.58

Jamal rides his bike at a uniform rate of 12 miles per hour for 312312 hours. What distance has he traveled?

Try It 2.115

Lindsay drove for 512512 hours at 60 miles per hour. How much distance did she travel?

Try It 2.116

Trinh walked for 213213 hours at 3 miles per hour. How far did she walk?

Example 2.59

Rey is planning to drive from his house in San Diego to visit his grandmother in Sacramento, a distance of 520 miles. If he can drive at a steady rate of 65 miles per hour, how many hours will the trip take?

Try It 2.117

Lee wants to drive from Phoenix to his brother’s apartment in San Francisco, a distance of 770 miles. If he drives at a steady rate of 70 miles per hour, how many hours will the trip take?

Try It 2.118

Yesenia is 168 miles from Chicago. If she needs to be in Chicago in 3 hours, at what rate does she need to drive?

Solve a Formula for a Specific Variable

You are probably familiar with some geometry formulas. A formula is a mathematical description of the relationship between variables. Formulas are also used in the sciences, such as chemistry, physics, and biology. In medicine they are used for calculations for dispensing medicine or determining body mass index. Spreadsheet programs rely on formulas to make calculations. It is important to be familiar with formulas and be able to manipulate them easily.

In Example 2.58 and Example 2.59, we used the formula d=rtd=rt. This formula gives the value of dd, distance, when you substitute in the values of randtrandt, the rate and time. But in Example 2.59, we had to find the value of tt. We substituted in values of dandrdandr and then used algebra to solve for tt. If you had to do this often, you might wonder why there is not a formula that gives the value of tt when you substitute in the values of dandrdandr. We can make a formula like this by solving the formula d=rtd=rt for tt.

To solve a formula for a specific variable means to isolate that variable on one side of the equals sign with a coefficient of 1. All other variables and constants are on the other side of the equals sign. To see how to solve a formula for a specific variable, we will start with the distance, rate and time formula.

Example 2.60

Solve the formula d=rtd=rt for tt:

  1. when d=520d=520 and r=65r=65
  2. in general
Try It 2.119

Solve the formula d=rtd=rt for rr:

when d=180andt=4d=180andt=4 in general

Try It 2.120

Solve the formula d=rtd=rt for rr:

when d=780andt=12d=780andt=12 in general

Example 2.61

Solve the formula A=12bhA=12bh for hh:

when A=90A=90 and b=15b=15 in general

Try It 2.121

Use the formula A=12bhA=12bh to solve for hh:

when A=170A=170 and b=17b=17 in general

Try It 2.122

Use the formula A=12bhA=12bh to solve for bb:

when A=62A=62 and h=31h=31 in general

The formula I=PrtI=Prt is used to calculate simple interest, I, for a principal, P, invested at rate, r, for t years.

Example 2.62

Solve the formula I=PrtI=Prt to find the principal, PP:

when I=$5,600,r=4%,t=7yearsI=$5,600,r=4%,t=7years in general

Try It 2.123

Use the formula I=PrtI=Prt to find the principal, PP:

when I=$2,160,r=6%,t=3yearsI=$2,160,r=6%,t=3years in general

Try It 2.124

Use the formula I=PrtI=Prt to find the principal,PP:

when I=$5,400,r=12%,t=5yearsI=$5,400,r=12%,t=5years in general

Later in this class, and in future algebra classes, you’ll encounter equations that relate two variables, usually x and y. You might be given an equation that is solved for y and need to solve it for x, or vice versa. In the following example, we’re given an equation with both x and y on the same side and we’ll solve it for y.

Example 2.63

Solve the formula 3x+2y=183x+2y=18 for y:

when x=4x=4 in general

Try It 2.125

Solve the formula 3x+4y=103x+4y=10 for y:

when x=143x=143 in general

Try It 2.126

Solve the formula 5x+2y=185x+2y=18 for y:

when x=4x=4 in general

In Examples 1.60 through 1.64 we used the numbers in part as a guide to solving in general in part . Now we will solve a formula in general without using numbers as a guide.

Example 2.64

Solve the formula P=a+b+cP=a+b+c for aa.

Try It 2.127

Solve the formula P=a+b+cP=a+b+c for b.

Try It 2.128

Solve the formula P=a+b+cP=a+b+c for c.

Example 2.65

Solve the formula 6x+5y=136x+5y=13 for y.

Try It 2.129

Solve the formula 4x+7y=94x+7y=9 for y.

Try It 2.130

Solve the formula 5x+8y=15x+8y=1 for y.

Section 2.6 Exercises

Practice Makes Perfect

Use the Distance, Rate, and Time Formula

In the following exercises, solve.

376.

Steve drove for 812812 hours at 72 miles per hour. How much distance did he travel?

377.

Socorro drove for 456456 hours at 60 miles per hour. How much distance did she travel?

378.

Yuki walked for 134134 hours at 4 miles per hour. How far did she walk?

379.

Francie rode her bike for 212212 hours at 12 miles per hour. How far did she ride?

380.

Connor wants to drive from Tucson to the Grand Canyon, a distance of 338 miles. If he drives at a steady rate of 52 miles per hour, how many hours will the trip take?

381.

Megan is taking the bus from New York City to Montreal. The distance is 380 miles and the bus travels at a steady rate of 76 miles per hour. How long will the bus ride be?

382.

Aurelia is driving from Miami to Orlando at a rate of 65 miles per hour. The distance is 235 miles. To the nearest tenth of an hour, how long will the trip take?

383.

Kareem wants to ride his bike from St. Louis to Champaign, Illinois. The distance is 180 miles. If he rides at a steady rate of 16 miles per hour, how many hours will the trip take?

384.

Javier is driving to Bangor, 240 miles away. If he needs to be in Bangor in 4 hours, at what rate does he need to drive?

385.

Alejandra is driving to Cincinnati, 450 miles away. If she wants to be there in 6 hours, at what rate does she need to drive?

386.

Aisha took the train from Spokane to Seattle. The distance is 280 miles and the trip took 3.5 hours. What was the speed of the train?

387.

Philip got a ride with a friend from Denver to Las Vegas, a distance of 750 miles. If the trip took 10 hours, how fast was the friend driving?

Solve a Formula for a Specific Variable

In the following exercises, use the formula d=rtd=rt.

388.

Solve for tt
when d=350d=350 and r=70r=70
in general

389.

Solve for tt
when d=240andr=60d=240andr=60
in general

390.

Solve for tt
when d=510andr=60d=510andr=60
in general

391.

Solve for tt
when d=175andr=50d=175andr=50
in general

392.

Solve for rr
when d=204andt=3d=204andt=3
in general

393.

Solve for rr
when d=420andt=6d=420andt=6
in general

394.

Solve for rr
when d=160andt=2.5d=160andt=2.5
in general

395.

Solve for rr
when d=180andt=4.5d=180andt=4.5
in general

In the following exercises, use the formula A=12bhA=12bh.

396.

Solve for bb
when A=126andh=18A=126andh=18
in general

397.

Solve for hh
when A=176andb=22A=176andb=22
in general

398.

Solve for hh
when A=375andb=25A=375andb=25
in general

399.

Solve for bb
when A=65andh=13A=65andh=13
in general

In the following exercises, use the formula I = Prt.

400.

Solve for the principal, P for
I=$5,480,r=4%,I=$5,480,r=4%,
t=7yearst=7years
in general

401.

Solve for the principal, P for
I=$3,950,r=6%,I=$3,950,r=6%,
t=5yearst=5years
in general

402.

Solve for the time, t for
I=$2,376,P=$9,000,I=$2,376,P=$9,000,
r=4.4%r=4.4%
in general

403.

Solve for the time, t for
I=$624,P=$6,000,I=$624,P=$6,000,
r=5.2%r=5.2%
in general

In the following exercises, solve.

404.

Solve the formula 2x+3y=122x+3y=12 for y
when x=3x=3
in general

405.

Solve the formula 5x+2y=105x+2y=10 for y
when x=4x=4
in general

406.

Solve the formula 3xy=73xy=7 for y
when x=−2x=−2
in general

407.

Solve the formula 4x+y=54x+y=5 for y
when x=−3x=−3
in general

408.

Solve a+b=90a+b=90 for bb.

409.

Solve a+b=90a+b=90 for aa.

410.

Solve 180=a+b+c180=a+b+c for aa.

411.

Solve 180=a+b+c180=a+b+c for cc.

412.

Solve the formula 8x+y=158x+y=15 for y.

413.

Solve the formula 9x+y=139x+y=13 for y.

414.

Solve the formula −4x+y=−6−4x+y=−6 for y.

415.

Solve the formula −5x+y=−1−5x+y=−1 for y.

416.

Solve the formula 4x+3y=74x+3y=7 for y.

417.

Solve the formula 3x+2y=113x+2y=11 for y.

418.

Solve the formula xy=−4xy=−4 for y.

419.

Solve the formula xy=−3xy=−3 for y.

420.

Solve the formula P=2L+2WP=2L+2W for LL.

421.

Solve the formula P=2L+2WP=2L+2W for WW.

422.

Solve the formula C=πdC=πd for dd.

423.

Solve the formula C=πdC=πd for ππ.

424.

Solve the formula V=LWHV=LWH for LL.

425.

Solve the formula V=LWHV=LWH for HH.

Everyday Math

426.

Converting temperature While on a tour in Greece, Tatyana saw that the temperature was 40o Celsius. Solve for F in the formula C=59(F32)C=59(F32) to find the Fahrenheit temperature.

427.

Converting temperature Yon was visiting the United States and he saw that the temperature in Seattle one day was 50o Fahrenheit. Solve for C in the formula F=95C+32F=95C+32 to find the Celsius temperature.

Writing Exercises

428.

Solve the equation 2x+3y=62x+3y=6 for yy
when x=−3x=−3
in general
Which solution is easier for you, or ? Why?

429.

Solve the equation 5x2y=105x2y=10 for xx
when y=10y=10
in general
Which solution is easier for you, or ? Why?

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 “use the distance, rate, and time formula,” and “solve a formula for a specific variable.” 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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