9.7 Solve a Formula for a Specific Variable

Learning objectives

Learning objectives

• Use the distance, rate, and time formula

• Solve a formula for a specific variable

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 $60$ miles per hour for $2$ hours? (This might happen if you use your car’s cruise control while driving on the Interstate.) If you said $120$ miles, you already know how to use this formula!

The math to calculate the distance might look like this:

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

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

Example 9.57

Solution

1. Identify the known quantities. Jamal travels at a uniform rate $r = 12$ miles per hour for $t = 3\frac{1}{2} = 3.5$ hours, and the distance $d$ is unknown.
2. Apply the distance formula. Substitute into $d = rt$ to obtain $d = 12(3.5)$ miles.
3. Multiply to find the distance. The product is $d = 42$ miles.
4. Check the reasonableness. The diagram lists $12$, $24$, $36$, and $48$ miles for $1$, $2$, $3$, and $4$ hours, so $42$ miles in $3.5$ hours lies between the corresponding values. Jamal traveled $42$ miles.

Takeaway: For a constant rate, multiply rate by elapsed time to find distance, and convert mixed numbers before multiplying.

Example 9.58

Solution

1. Identify the known quantities. The distance is $d=520$ miles, the rate is $r=65$ miles per hour, and the unknown time is $t$ in hours.
2. Apply the distance formula. Substitute the values into $d=rt$ to write $520=65t$.
3. Solve for time. Divide both sides by $65$ to obtain $t=520/65=8$ hours. The units also reduce from miles per miles per hour to hours.
4. Check the travel time. Multiplying the rate by the time gives $65(8)=520$ miles, which agrees with the stated distance.
5. State the trip duration. Rey's trip will take $8$ hours.

Takeaway: Use the distance formula $d=rt$ and divide distance by rate, so $t=520/65=8$ hours.

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

To solve a formula for a specific variable means to get that variable by itself with a coefficient of $1$ 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

Solution

1. Substitute the given values. With $d=520$ and $r=65$, the formula $d=rt$ becomes $520=65t$, matching the numerical work in the figures.
2. Solve the numerical equation. Divide both sides by $65$ to get $t=520/65=8$, so the time is $8$ hours when the distance is $520$ miles and the rate is $65$ miles per hour.
3. Solve the formula for t. In the general equation $d=rt$, divide every term by $r$ to obtain $\frac{d}{r}=t$, and write the equivalent form $t=\frac{d}{r}$.
4. Check the solved formula. Multiplying $\frac{d}{r}$ by $r$ gives $d$, so the rearranged formula remains equivalent to the original distance formula.
5. State both results. For the given values $t=8$ hours, and in general the formula solved for $t$ is $t=\frac{d}{r}$.

Takeaway: Solving $d=rt$ for time gives $t=d/r$, the formula for finding travel time whenever distance and rate are known.

Example 9.60

Solution

1. Substitute the known triangle measurements. Starting with $A = \frac{1}{2}bh$, use $A = 90$ and $b = 15$ to write $90 = \frac{1}{2}(15)h$.
2. Remove the fraction from both sides. Multiply every term by $2$ to obtain $180 = 15h$, as shown by the highlighted $2$ in the figures.
3. Isolate the height. Divide both sides by $15$ to get $h = 12$.
4. Solve the general formula. Multiplying $A = \frac{1}{2}bh$ by $2$ gives $2A = bh$, and dividing by $b$ gives $h = \frac{2A}{b}$. Thus the height is $12$ when $A = 90$ and $b = 15$, and in general $h = \frac{2A}{b}$.

Takeaway: To isolate a variable, undo the operation in the formula on both sides while preserving equality at every step.

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

Example 9.61

Solution

1. Write the interest formula. The formula $I = Prt$ relates interest $I$, principal $P$, rate $r$, and time $t$, and the goal is to isolate $P$.
2. Substitute the known values. With $I = 5{,}600$, $r = 0.04$, and $t = 7$, write $5{,}600 = P(0.04)(7) = 0.28P$.
3. Solve for the principal. Divide both sides by $0.28$ to obtain $P = \frac{5{,}600}{0.28} = 20{,}000$.
4. Solve in general. Divide every term in $I = Prt$ by $rt$, giving $P = \frac{I}{rt}$. The figures show the same cancellation of the combined factor $0.28$; therefore the principal is $20{,}000$ dollars, and in general $P = \frac{I}{rt}$.

Takeaway: In simple interest, the principal is the interest divided by the product of the decimal rate and time.

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

Example 9.62

Solution

1. Substitute the given value of x. In $3x + 2y = 18$, setting $x = 4$ gives $3(4) + 2y = 18$, or $12 + 2y = 18$.
2. Isolate the y term. Subtract $12$ from both sides to obtain $2y = 6$.
3. Solve for y. Divide both sides by $2$ to get $y = 3$.
4. Solve the equation in general. Subtract $3x$ from both sides of $3x + 2y = 18$ to get $2y = 18 - 3x$, then divide by $2$ to obtain $y = \frac{18 - 3x}{2}$. Thus $y = 3$ when $x = 4$, and the general solution is $y = \frac{18 - 3x}{2}$.

Takeaway: When solving for a variable in a two-variable equation, remove the other variable's term before dividing by the coefficient.

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

Solution

1. Identify the target term. In $P = a + b + c$, the goal is to isolate $a$ on one side of the equation.
2. Subtract the other terms from both sides. This gives $P - b - c = a + b + c - b - c$.
3. Simplify both sides. The terms $b + c$ cancel on the right, leaving $P - b - c = a$.
4. Write the solved formula. Interchanging the equivalent sides gives $a = P - b - c$. Thus the formula solved for $a$ is $a = P - b - c$.

Takeaway: To isolate a term that is added to others, subtract those other terms from both sides and simplify.

Example 9.64

Solution

1. Locate the term attached to y. In $3x + y = 10$, the term $3x$ must be removed from the left side.
2. Subtract $3x$ from both sides. This produces $3x - 3x + y = 10 - 3x$, matching the displayed figure.
3. Simplify the left side. The terms $3x - 3x$ equal zero, leaving $y = 10 - 3x$.
4. State the solution. The equation solved for $y$ is $y = 10 - 3x$.

Takeaway: Subtracting a term from both sides preserves equality and isolates the requested variable.

Example 9.65

Solution

1. Subtract the x term from both sides. Starting with $6x + 5y = 13$, write $6x + 5y - 6x = 13 - 6x$, as shown in the second figure.
2. Simplify the left side. Since $6x - 6x = 0$, this reduces to $5y = 13 - 6x$.
3. Divide both sides by 5. The coefficient $5$ cancels to give $\frac{5y}{5} = \frac{13 - 6x}{5}$.
4. State the solution. Thus $y = \frac{13 - 6x}{5}$, which is the equation solved for $y$.

Takeaway: Remove terms containing other variables first, then divide by the numerical coefficient of the target variable.

Section 9.7 Exercises

Section 9.7 Exercises

Use the Distance, Rate, and Time Formula

In the following exercises, solve.

Solve a Formula for a Specific Variable

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

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

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

In the following exercises, solve.

Everyday Math

Writing Exercises

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?

1. Lindsay drove for $5\frac{1}{2}$ hours at $60$ miles per hour. How much distance did she travel?

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

3. 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?

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

2. Solve the formula $d=rt$ for $r:$

1. ⓐ when $d=180$ and $t=4$

2. ⓐ in general

3. Solve the formula $d=rt$ for $r:$

1. ⓐ when $d=780$ and $t=12$

2. ⓑ in general

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

1. ⓐ when $A=170$ and $b=17$

2. ⓑ in general

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

1. ⓐ when $A=62$ and $h=31$

2. ⓑ in general

3. Use the formula $I=Prt.$

Find $t:$ ⓐ when $I=\text{\textdollar 2,160},\,r=\mathrm{6\%},\,P=\text{\textdollar 12,000;}$ ⓑ in general

1. Use the formula $I=Prt.$

Find $r:$ ⓐ when $I=\text{\textdollar 5,400},\,P=\text{\textdollar 9,000},\,t=5\,\mathrm{years}$ ⓑ in general

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

1. ⓐ when $x=2$

2. ⓑ in general

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

1. ⓐ when $x=4$

2. ⓑ in general

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

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

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

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

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

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

1. Steve drove for $8\frac{1}{2}$ hours at $72$ miles per hour. How much distance did he travel?

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

3. 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?

1. 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?

2. 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?

3. 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?

1. Solve for $t:$

1. ⓐ when $d=350$ and $r=70$

2. ⓑ in general

2. Solve for $t:$

1. ⓐ when $d=510$ and $r=60$

2. ⓑ in general

3. Solve for $r:$

1. ⓐ when $d=204$ and $t=3$

2. ⓑ in general

1. Solve for $r:$

1. ⓐ when $d=160$ and $t=2.5$

2. ⓑ in general

2. Solve for $b:$

1. ⓐ when $A=126$ and $h=18$

2. ⓑ in general

3. Solve for $h:$

1. ⓐ when $A=375$ and $b=25$

2. ⓑ in general

1. Solve for the principal, $P$ for:

1. ⓐ $\,I=\text{\textdollar 5,480}$, $r=\mathrm{4\%}$, $t=7\,\mathrm{years}$

2. ⓑ in general

2. Solve for the time, $t$ for:

1. ⓐ $\,I=\text{\textdollar 2,376}$, $P=\text{\textdollar 9,000}$, $r=4.4\%$

2. ⓑ in general

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

1. ⓐ when $x=3$

2. ⓑ in general

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

1. ⓐ when $x=-2$

2. ⓑ in general

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

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

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

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

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

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

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

3. Solve the formula $C=\pi{}d$ for $d.$

1. Solve the formula $V=LWH$ for $L.$

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

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

1. ⓐ when $x=-3$

2. ⓑ in general

3. ⓒ Which solution is easier for you? Explain why.