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# 9.7: Solve a Formula for a Specific Variable

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Skills to Develop

• Use the distance, rate, and time formula
• Solve a formula for a specific variable

Be Prepared!

Before you get started, take this readiness quiz.

1. Write 35 miles per gallon as a unit rate. If you missed this problem, review Example 5.65.
2. Solve 6x + 24 = 96. If you missed this problem, review Example 8.20.

##### Solution
 (a) when I = $5,600, r = 4%, t = 7 years (b) in general Write the forumla. I = Prt I = Prt Substitute any given values. 5600 = P(0.04)(7) I = Prt Multiply r • t. 5600 = P(0.28) I = P(rt) Divide to isolate P. $$\frac{5600}{\textcolor{red}{0.28}} = \frac{P(0.28)}{\textcolor{red}{0.28}}$$ $$\frac{I}{\textcolor{red}{rt}} = \frac{P(rt)}{\textcolor{red}{rt}}$$ Simplify. 20,000 = P $$\frac{I}{rt}$$ = P State the answer. The principal is$20,000. $$P = \frac{I}{rt}$$

Exercise 9.121:

Use the formula I = Prt. Find t: (a) when I = $2,160, r = 6%, P =$12,000; (b) in general

Exercise 9.122:

Use the formula I = Prt. Find r: (a) when I = $5,400, P =$9,000, t = 5 years; (b) in general

Later in this course, 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. To do this, we will follow the same steps that we used to solve a formula for a specific variable.

Example 9.62:

Solve the formula 3x + 2y = 18 for y: (a) when x = 4 (b) in general

##### Solution
 (a) when x = 4 (b) in general Write the equation. 3x + 2y = 18 3x + 2y = 18 Substitute any given values. 3(4) + 2y = 18 3x + 2y = 18 Simplify if possible. 12 + 2y = 18 3x + 2y = 18 Subtract to isolate the y-term. $$12 \textcolor{red}{-12} + 2y = 18 \textcolor{red}{-12}$$ $$3x \textcolor{red}{-3x}+ 2y = 18 \textcolor{red}{-3x}$$ Simplify. 2y = 6 2y = 18 - 3x Divide. $$\frac{2y}{\textcolor{red}{2}} = \frac{6}{\textcolor{red}{2}}$$ $$\frac{2y}{\textcolor{red}{2}} = \frac{18 - 3x}{\textcolor{red}{2}}$$ Simplify. y = 3 $$y = \frac{18 - 3x}{2}$$

Exercise 9.123:

Solve the formula 3x + 4y = 10 for y: (a) when x = 2 (b) in general

Exercise 9.124:

Solve the formula 5x + 2y = 18 for y: (a) when x = 4 (b) in general

In the previous examples, we used the numbers in part (a) as a guide to solving in general in part (b). Do you think you’re ready to solve a formula in general without using numbers as a guide?

Example 9.63:

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

##### Solution

We will isolate a on one side of the equation.

 We will isolate a on one side of the equation. Write the equation. P = a + b + c Subtract b and c from both sides to isolate a. $$P \textcolor{red}{-b -c} = a + b + c \textcolor{red}{-b -c}$$ Simplify. P − b − c = a

So, a = P − b − c.

Exercise 9.125:

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

Exercise 9.126:

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

Example 9.64:

Solve the equation 3x + y = 10 for y.

##### Solution

We will isolate y on one side of the equation.

 We will isolate y on one side of the equation. Write the equation. 3x + y = 10 Subtract 3x from both sides to isolate y. $$3x \textcolor{red}{-3x} + y= 10 \textcolor{red}{-3x}$$ Simplify. y = 10 - 3x

Exercise 9.127:

Solve the formula 7x + y = 11 for y

Exercise 9.128:

Solve the formula 11x + y = 8 for y.

Example 9.65:

Solve the equation 6x + 5y = 13 for y.

##### Solution

We will isolate y on one side of the equation.

 We will isolate y on one side of the equation. Write the equation. 6x + 5y = 13 Subtract to isolate the term with y. $$6x + 5y \textcolor{red}{-6x} = 13 \textcolor{red}{-6x}$$ Simplify. 5y = 13 - 6x Divide 5 to make the coefficient 1. $$\frac{5y}{\textcolor{red}{5}} = \frac{13 - 6x}{\textcolor{red}{5}}$$ Simplify. $$y = \frac{13 - 6x}{5}$$

Exercise 9.129:

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

Exercise 9.130:

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

### Practice Makes Perfect

#### Use the Distance, Rate, and Time Formula

In the following exercises, solve.

1. Steve drove for $$8 \frac{1}{2}$$ hours at 72 miles per hour. How much distance did he travel?
2. Socorro drove for $$4 \frac{5}{6}$$ hours at 60 miles per hour. How much distance did she travel?
3. Yuki walked for $$1 \frac{3}{4}$$ hours at 4 miles per hour. How far did she walk?
4. Francie rode her bike for $$2 \frac{1}{2}$$ hours at 12 miles per hour. How far did she ride?
5. 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?
6. Megan is taking the bus from New York City to Montreal. The distance is 384 miles and the bus travels at a steady rate of 64 miles per hour. How long will the bus ride be?
7. 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?
8. Kareem wants to ride his bike from St. Louis, Missouri 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?
9. Javier is driving to Bangor, Maine, which is 240 miles away from his current location. If he needs to be in Bangor in 4 hours, at what rate does he need to drive?
10. Alejandra is driving to Cincinnati, Ohio, 450 miles away. If she wants to be there in 6 hours, at what rate does she need to drive?
11. 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?
12. 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 = rt.

1. Solve for t: (a) when d = 350 and r = 70 (b) in general
2. Solve for t: (a) when d = 240 and r = 60 (b) in general
3. Solve for t: (a) when d = 510 and r = 60 (b) in general
4. Solve for t: (a) when d = 175 and r = 50 (b) in general
5. Solve for r: (a) when d = 204 and t = 3 (b) in general
6. Solve for r: (a) when d = 420 and t = 6 (b) in general
7. Solve for r: (a) when d = 160 and t = 2.5 (b) in general
8. Solve for r: (a) when d = 180 and t = 4.5 (b) in general.

In the following exercises, use the formula A = $$\frac{1}{2}$$bh.

1. Solve for b: (a) when A = 126 and h = 18 (b) in general
2. Solve for h: (a) when A = 176 and b = 22 (b) in general
3. Solve for h: (a) when A = 375 and b = 25 (b) in general
4. Solve for b: (a) when A = 65 and h = 13 (b) in general

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

1. Solve for the principal, P for: (a) I = $5,480, r = 4%, t = 7 years (b) in general 2. Solve for the principal, P for: (a) I =$3,950, r = 6%, t = 5 years (b) in general
3. Solve for the time, t for: (a) I = $2,376, P =$9,000, r = 4.4% (b) in general
4. Solve for the time, t for: (a) I = $624, P =$6,000, r = 5.2% (b) in general

In the following exercises, solve.

1. Solve the formula 2x + 3y = 12 for y: (a) when x = 3 (b) in general
2. Solve the formula 5x + 2y = 10 for y: (a) when x = 4 (b) in general
3. Solve the formula 3x + y = 7 for y: (a) when x = −2 (b) in general
4. Solve the formula 4x + y = 5 for y: (a) when x = −3 (b) in general
5. Solve a + b = 90 for b.
6. Solve a + b = 90 for a.
7. Solve 180 = a + b + c for a.
8. Solve 180 = a + b + c for c.
9. Solve the formula 8x + y = 15 for y.
10. Solve the formula 9x + y = 13 for y.
11. Solve the formula −4x + y = −6 for y.
12. Solve the formula −5x + y = −1 for y.
13. Solve the formula 4x + 3y = 7 for y.
14. Solve the formula 3x + 2y = 11 for y.
15. Solve the formula x − y = −4 for y.
16. Solve the formula x − y = −3 for y.
17. Solve the formula P = 2L + 2W for L.
18. Solve the formula P = 2L + 2W for W.
19. Solve the formula C = $$\pi$$d for d.
20. Solve the formula C = $$\pi$$d for $$\pi$$.
21. Solve the formula V = LWH for L.
22. Solve the formula V = LWH for H.

### Everyday Math

1. Converting temperature While on a tour in Greece, Tatyana saw that the temperature was 40° Celsius. Solve for F in the formula C = $$\frac{5}{9}$$(F − 32) to find the temperature in Fahrenheit.
2. Converting temperature Yon was visiting the United States and he saw that the temperature in Seattle was 50° Fahrenheit. Solve for C in the formula F = $$\frac{9}{5}$$C + 32 to find the temperature in Celsius.

### Writing Exercises

1. Solve the equation 2x + 3y = 6 for y: (a) when x = −3 (b) in general (c) Which solution is easier for you? Explain why.
2. Solve the equation 5x − 2y = 10 for x: (a) when y = 10 (b) in general (c) Which solution is easier for you? Explain why.

### Self Check

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

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