Contents

Solve a Formula for a Specific Variable

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 9.7

Before you get started, take this readiness quiz.

Write $3535$ miles per gallon as a unit rate.
If you missed this problem, review Example 5.65.
Solve $6 x + 24 = 96 . 6 x + 24 = 96 .$
If you missed this problem, review Example 8.20.
Find the simple interest earned after $55$ years on $$1,000 $1,000$ at an interest rate of $4% . 4% .$
If you missed this problem, review Example 6.33.

Use the Distance, Rate, and Time Formula

One formula you’ll use often in algebra and in everyday life is the formula for distance traveled by an object moving at a constant speed. The basic idea is probably already familiar to you. Do you know what distance you travel if you drove at a steady rate of $6060$ miles per hour for $22$ hours? (This might happen if you use your car’s cruise control while driving on the Interstate.) If you said $120120$ miles, you already know how to use this formula!

The math to calculate the distance might look like this:

\begin{matrix} distance = (\frac{60 miles}{1 hour}) (2 hours) \\ distance = 120 miles \end{matrix} \begin{matrix} distance = (\frac{60}{1}) (2) & distance = 120 \end{matrix}

In general, the formula relating distance, rate, and time is

distance = rate \cdot time distance

Distance, Rate and Time

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

d = r t d = r t

where $d = d =$ distance, $r = r =$ rate, and $t = t =$ time.

Notice that the units we used above for the rate were miles per hour, which we can write as a ratio $\frac{m i l e s}{h o u r} . \frac{m i l e s}{h o u r} .$ Then when we multiplied by the time, in hours, the common units ‘hour’ divided out. The answer was in miles.

Example 9.57

Jamal rides his bike at a uniform rate of $1212$ miles per hour for $3 \frac{1}{2} 3 \frac{1}{2}$ hours. How much distance has he traveled?

Show solution

Solution

Step 1. Read the problem. You may want to create a mini-chart to summarize the information in the problem.	$d = ? d = ?$ $r = 12 mph r = 12$ $t = 3 \frac{1}{2} hours t = 3 \frac{1}{2}$
Step 2. Identify what you are looking for.	distance traveled
Step 3. Name. Choose a variable to represent it.	let d = distance
Step 4. Translate. Write the appropriate formula for the situation. Substitute in the given information.	$d = r t d = r t$ $d = 12 \cdot 3 \frac{1}{2} d = 12 \cdot 3 \frac{1}{2}$
Step 5. Solve the equation.	$d = 42 miles d = 42$
Step 6. Check: Does 42 miles make sense?
Step 7. Answer the question with a complete sentence.	Jamal rode 42 miles.

Try It 9.113

Lindsay drove for $5 \frac{1}{2} 5 \frac{1}{2}$ hours at $6060$ miles per hour. How much distance did she travel?

Try It 9.114

Trinh walked for $2 \frac{1}{3} 2 \frac{1}{3}$ hours at $33$ miles per hour. How far did she walk?

Example 9.58

Rey is planning to drive from his house in San Diego to visit his grandmother in Sacramento, a distance of $520520$ miles. If he can drive at a steady rate of $6565$ miles per hour, how many hours will the trip take?

Show solution

Solution

Step 1. Read the problem. Summarize the information in the problem.	$d = 520 miles d = 520$ $r = 65 mph r = 65$ $t = ? t = ?$
Step 2. Identify what you are looking for.	how many hours (time)
Step 3. Name: Choose a variable to represent it.	let t = time
Step 4. Translate. Write the appropriate formula. Substitute in the given information.	$d = r t d = r t$ $520 = 65 t 520 = 65 t$
Step 5. Solve the equation.	$t = 8 t = 8$
Step 6. Check: Substitute the numbers into the formula and make sure the result is a true statement. $d = r t d = r t$ $520 \overset{?}{=} 65 \cdot 8 520 \overset{?}{=} 65 \cdot 8$ $520 = 520> ✓ 520 = 520> ✓$
Step 7. Answer the question with a complete sentence. We know the units of time will be hours because we divided miles by miles per hour.	Rey's trip will take 8 hours.

Try It 9.115

Lee wants to drive from Phoenix to his brother’s apartment in San Francisco, a distance of $770770$ miles. If he drives at a steady rate of $7070$ miles per hour, how many hours will the trip take?

Try It 9.116

Yesenia is $168168$ miles from Chicago. If she needs to be in Chicago in $33$ hours, at what rate does she need to drive?

Solve a Formula for a Specific Variable

In this chapter, you became familiar with some formulas used in geometry. Formulas are also very useful in the sciences and social sciences—fields such as chemistry, physics, biology, psychology, sociology, and criminal justice. Healthcare workers use formulas, too, even for something as routine as dispensing medicine. The widely used spreadsheet program Microsoft Excel^TM relies on formulas to do its calculations. Many teachers use spreadsheets to apply formulas to compute student grades. It is important to be familiar with formulas and be able to manipulate them easily.

In Example 9.57 and Example 9.58, we used the formula $d = r t . d = r t .$ This formula gives the value of $d d$ when you substitute in the values of $r r$ and $t . t .$ But in Example 9.58, we had to find the value of $t . t .$ We substituted in values of $d d$ and $r r$ and then used algebra to solve to $t . t .$ If you had to do this often, you might wonder why there isn’t a formula that gives the value of $t t$ when you substitute in the values of $d d$ and $r . r .$ We can get a formula like this by solving the formula $d = r t d = r t$ for $t . t .$

To solve a formula for a specific variable means to get that variable by itself with a coefficient of $11$ on one side of the equation and all the other variables and constants on the other side. We will call this solving an equation for a specific variable in general. This process is also called solving a literal equation. The result is another formula, made up only of variables. The formula contains letters, or literals.

Let’s try a few examples, starting with the distance, rate, and time formula we used above.

Example 9.59

Solve the formula $d = r t d = r t$ for $t : t :$

ⓐ when $d = 520 d = 520$ and $r = 65 r = 65$
ⓑ in general.

Show solution

Solution

We’ll write the solutions side-by-side so you can see that solving a formula in general uses the same steps as when we have numbers to substitute.

	ⓐ when d = 520 and r = 65	ⓑ in general
Write the forumla.
Substitute any given values.
Divide to isolate t.
Simplify.

Notice that the solution for ⓐ is the same as that in Example 9.58. We say the formula $t = \frac{d}{r} t = \frac{d}{r}$ is solved for $t . t .$ We can use this version of the formula anytime we are given the distance and rate and need to find the time.

Try It 9.117

Solve the formula $d = r t d = r t$ for $r : r :$

ⓐ when $d = 180 d = 180$ and $t = 4 t = 4$
ⓐ in general

Try It 9.118

Solve the formula $d = r t d = r t$ for $r : r :$

ⓐ when $d = 780 d = 780$ and $t = 12 t = 12$
ⓑ in general

We used the formula $A = \frac{1}{2} b h A = \frac{1}{2} b h$ in Use Properties of Rectangles, Triangles, and Trapezoids to find the area of a triangle when we were given the base and height. In the next example, we will solve this formula for the height.

Example 9.60

The formula for area of a triangle is $A = \frac{1}{2} b h . A = \frac{1}{2} b h .$ Solve this formula for $h : h :$

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

Show solution

Solution

	ⓐ when A = 90 and b = 15	ⓑ in general
Write the forumla.
Substitute any given values.
Clear the fractions.
Simplify.
Solve for h.

We can now find the height of a triangle, if we know the area and the base, by using the formula

h = \frac{2 A}{b} h = \frac{2 A}{b}

Try It 9.119

Use the formula $A = \frac{1}{2} b h A = \frac{1}{2} b h$ to solve for $h: h:$

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

Try It 9.120

Use the formula $A = \frac{1}{2} b h A = \frac{1}{2} b h$ to solve for $b : b :$

ⓐ when $A = 62 A = 62$ and $h = 31 h = 31$
ⓑ in general

In Solve Simple Interest Applications, we used the formula $I = P r t I = P r t$ to calculate simple interest, where $I I$ is interest, $P P$ is principal, $r r$ is rate as a decimal, and $t t$ is time in years.

Example 9.61

Solve the formula $I = P r t I = P r t$ to find the principal, $P : P :$

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

Show solution

Solution

	I = $5600, r = 4%, t = 7 years	in general
Write the forumla.
Substitute any given values.
Multiply r ⋅ t.
Divide to isolate P.
Simplify.
State the answer.	The principal is $20,000.

Try It 9.121

Use the formula $I = P r t . I = P r t .$

Find $t : t :$ ⓐ when $I = $2,160, r = 6%, P = $12,000; I = $2,160,$ ⓑ in general

Try It 9.122

Use the formula $I = P r t . I = P r t .$

Find $r : r :$ ⓐ when $I = $5,400, P = $9,000, t = 5 years I = $5,400,$ ⓑ in general

Later in this class, and in future algebra classes, you’ll encounter equations that relate two variables, usually $x x$ and $y . y .$ You might be given an equation that is solved for $y y$ and need to solve it for $x, x,$ or vice versa. In the following example, we’re given an equation with both $x x$ and $y y$ on the same side and we’ll solve it for $y . y .$ 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 $3 x + 2 y = 18 3 x + 2 y = 18$ for $y : y :$

ⓐ when $x = 4 x = 4$
ⓑ in general

Show solution

Solution

	when x = 4	in general
Write the equation.
Substitute any given values.
Simplify if possible.
Subtract to isolate the y-term.
Simplify.
Divide.
Simplify.

Try It 9.123

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

ⓐ when $x = 2 x = 2$
ⓑ in general

Try It 9.124

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

ⓐ when $x = 4 x = 4$
ⓑ 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 $P = a + b + c P = a + b + c$ for $a . a .$

Show solution

Solution

We will isolate $a a$ on one side of the equation.

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

So, $a = P - b - c a = P - b - c$

Try It 9.125

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

Try It 9.126

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

Example 9.64

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

Show solution

Solution

We will isolate $y y$ on one side of the equation.

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

Try It 9.127

Solve the formula $7 x + y = 11 7 x + y = 11$ for $y . y .$

Try It 9.128

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

Example 9.65

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

Show solution

Solution

We will isolate $y y$ on one side of the equation.

We will isolate y on one side of the equation.
Write the equation.
Subtract to isolate the term with y.
Simplify.
Divide 5 to make the coefficient 1.
Simplify.

Try It 9.129

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

Try It 9.130

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

Links To Literacy

The Links to Literacy activity What's Faster than a Speeding Cheetah? will provide you with another view of the topics covered in this section.

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Practice Makes Perfect

Use the Distance, Rate, and Time Formula

In the following exercises, solve.

307.

Steve drove for $8 \frac{1}{2} 8 \frac{1}{2}$ hours at $7272$ miles per hour. How much distance did he travel?

308.

Socorro drove for $4 \frac{5}{6} 4 \frac{5}{6}$ hours at $6060$ miles per hour. How much distance did she travel?

309.

Yuki walked for $1 \frac{3}{4} 1 \frac{3}{4}$ hours at $44$ miles per hour. How far did she walk?

310.

Francie rode her bike for $2 \frac{1}{2} 2 \frac{1}{2}$ hours at $1212$ miles per hour. How far did she ride?

311.

Connor wants to drive from Tucson to the Grand Canyon, a distance of $338338$ miles. If he drives at a steady rate of $5252$ miles per hour, how many hours will the trip take?

312.

Megan is taking the bus from New York City to Montreal. The distance is $384384$ miles and the bus travels at a steady rate of $6464$ miles per hour. How long will the bus ride be?

313.

Aurelia is driving from Miami to Orlando at a rate of $6565$ miles per hour. The distance is $235235$ miles. To the nearest tenth of an hour, how long will the trip take?

314.

Kareem wants to ride his bike from St. Louis, Missouri to Champaign, Illinois. The distance is $180180$ miles. If he rides at a steady rate of $1616$ miles per hour, how many hours will the trip take?

315.

Javier is driving to Bangor, Maine, which is $240240$ miles away from his current location. If he needs to be in Bangor in $44$ hours, at what rate does he need to drive?

316.

Alejandra is driving to Cincinnati, Ohio, $450450$ miles away. If she wants to be there in $66$ hours, at what rate does she need to drive?

317.

Aisha took the train from Spokane to Seattle. The distance is $280280$ miles, and the trip took $3.5 3.5$ hours. What was the speed of the train?

318.

Philip got a ride with a friend from Denver to Las Vegas, a distance of $750750$ miles. If the trip took $1010$ hours, how fast was the friend driving?

Solve a Formula for a Specific Variable

In the following exercises, use the formula. $d = r t . d = r t .$

319.

Solve for $t : t :$

ⓐ when $d = 350 d = 350$ and $r = 70 r = 70$
ⓑ in general

320.

Solve for $t : t :$

ⓐ when $d = 240 d = 240$ and $r = 60 r = 60$
ⓑ in general

321.

Solve for $t : t :$

ⓐ when $d = 510 d = 510$ and $r = 60 r = 60$
ⓑ in general

322.

Solve for $t : t :$

ⓐ when $d = 175 d = 175$ and $r = 50 r = 50$
ⓑ in general

323.

Solve for $r : r :$

ⓐ when $d = 204 d = 204$ and $t = 3 t = 3$
ⓑ in general

324.

Solve for $r : r :$

ⓐ when $d = 420 d = 420$ and $t = 6 t = 6$
ⓑ in general

325.

Solve for $r : r :$

ⓐ when $d = 160 d = 160$ and $t = 2.5 t = 2.5$
ⓑ in general

326.

Solve for $r : r :$

ⓐ when $d = 180 d = 180$ and $t = 4.5 t = 4.5$
ⓑ in general.

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

327.

Solve for $b : b :$

ⓐ when $A = 126 A = 126$ and $h = 18 h = 18$
ⓑ in general

328.

Solve for $h : h :$

ⓐ when $A = 176 A = 176$ and $b = 22 b = 22$
ⓑ in general

329.

Solve for $h : h :$

ⓐ when $A = 375 A = 375$ and $b = 25 b = 25$
ⓑ in general

330.

Solve for $b : b :$

ⓐ when $A = 65 A = 65$ and $h = 13 h = 13$
ⓑ in general

In the following exercises, use the formula $I = P r t . I = P r t .$

331.

Solve for the principal, $P P$ for:

ⓐ $I = $5,480$ , $r = 4% r = 4%$ , $t = 7 years t = 7$
ⓑ in general

332.

Solve for the principal, $P P$ for:

ⓐ $I = $3,950$ , $r = 6% r = 6%$ , $t = 5 years t = 5$
ⓑ in general

333.

Solve for the time, $t t$ for:

ⓐ $I = $2,376$ , $P = $9,000 P = $9,000$ , $r = 4.4 % r = 4.4 %$
ⓑ in general

334.

Solve for the time, $t t$ for:

ⓐ $I = $624$ , $P = $6,000 P = $6,000$ , $r = 5.2 % r = 5.2 %$
ⓑ in general

In the following exercises, solve.

335.

Solve the formula $2 x + 3 y = 12 2 x + 3 y = 12$ for $y : y :$

ⓐ when $x = 3 x = 3$
ⓑ in general

336.

Solve the formula $5 x + 2 y = 10 5 x + 2 y = 10$ for $y : y :$

ⓐ when $x = 4 x = 4$
ⓑ in general

337.

Solve the formula $3 x + y = 7 3 x + y = 7$ for $y : y :$

ⓐ when $x = −2 x = −2$
ⓑ in general

338.

Solve the formula $4 x + y = 5 4 x + y = 5$ for $y : y :$

ⓐ when $x = −3 x = −3$
ⓑ in general

339.

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

340.

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

341.

Solve $180 = a + b + c 180 = a + b + c$ for $a . a .$

342.

Solve $180 = a + b + c 180 = a + b + c$ for $c . c .$

343.

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

344.

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

345.

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

346.

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

347.

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

348.

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

349.

Solve the formula $x - y = −4 x - y = −4$ for $y . y .$

350.

Solve the formula $x - y = −3 x - y = −3$ for $y . y .$

351.

Solve the formula $P = 2 L + 2 W P = 2 L + 2 W$ for $L . L .$

352.

Solve the formula $P = 2 L + 2 W P = 2 L + 2 W$ for $W . W .$

353.

Solve the formula $C = π d C = π d$ for $d . d .$

354.

Solve the formula $C = π d C = π d$ for $π . π .$

355.

Solve the formula $V = L W H V = L W H$ for $L . L .$

356.

Solve the formula $V = L W H V = L W H$ for $H . H .$

Everyday Math

357.

Converting temperature While on a tour in Greece, Tatyana saw that the temperature was $40° 40°$ Celsius. Solve for $F F$ in the formula $C = \frac{5}{9} (F - 32) C = \frac{5}{9} (F - 32)$ to find the temperature in Fahrenheit.

358.

Converting temperature Yon was visiting the United States and he saw that the temperature in Seattle was $50° 50°$ Fahrenheit. Solve for $C C$ in the formula $F = \frac{9}{5} C + 32 F = \frac{9}{5} C + 32$ to find the temperature in Celsius.

Writing Exercises

359.

Solve the equation $2 x + 3 y = 6 2 x + 3 y = 6$ for $y : y :$

ⓐ when $x = −3 x = −3$
ⓑ in general

360.

Solve the equation $5 x - 2 y = 10 5 x - 2 y = 10$ for $x : x :$

ⓐ when $y = 10 y = 10$
ⓑ in general

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?