2.6 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 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!

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.

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

Solution

Example 2.59

Solution

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 $d=rt$. This formula gives the value of $d$, distance, when you substitute in the values of $r\,\mathrm{and}\,t$, the rate and time. But in Example 2.59, we had to find the value of $t$. We substituted in values of $d\,\mathrm{and}\,r$ and then used algebra to solve for $t$. If you had to do this often, you might wonder why there is not a formula that gives the value of $t$ when you substitute in the values of $d\,\mathrm{and}\,r$. We can make a formula like this by solving the formula $d=rt$ for $t$.

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

Solution

We will write the solutions side-by-side to demonstrate that solving a formula in general uses the same steps as when we have numbers to substitute.

We say the formula $t=\frac{d}{r}$ is solved for $t$.

Example 2.61

Solution

We can now find the height of a triangle, if we know the area and the base, by using the formula $h=\frac{2A}{b}$.

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

Example 2.62

Solution

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

Solution

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

Solution

Example 2.65

Solution

The fraction is simplified. We cannot divide $13-6x$ by 5.

Section 2.6 Exercises

Section 2.6 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

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

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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 $r$:

ⓐ when $d=180\,\mathrm{and}\,t=4$ ⓑ in general

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

ⓐ when $d=780\,\mathrm{and}\,t=12$ ⓑ in general

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

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

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

ⓐ when $A=62$ and $h=31$ ⓑ in general

3. Use the formula $I=Prt$ to find the principal, $P$:

ⓐ when $I=$2,160,r=6\%,t=3\,\mathrm{years}\,$ ⓑ in general

1. Use the formula $I=Prt$ to find the principal,$P$:

ⓐ when $I=$5,400,r=12\%,t=5\,\mathrm{years}\,$ ⓑ in general

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

ⓐ when $x=\frac{14}{3}$ ⓑ in general

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

ⓐ when $x=4$ ⓑ 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 $4x+7y=9$ for y.

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

2. Socorro drove for $4\frac{5}{6}$ hours at 60 miles per hour. How much distance did she travel?

3. Francie rode her bike for $2\frac{1}{2}$ hours at 12 miles per hour. How far did she ride?

1. 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?

2. 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?

3. 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?

1. 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?

2. Solve for $t$ ⓐ when $d=240\,\mathrm{and}\,r=60$ ⓑ in general

3. Solve for $t$ⓐ when $d=175\,\mathrm{and}\,r=50$ⓑ in general

1. Solve for $r$ⓐ when $d=420\,\mathrm{and}\,t=6$ⓑ in general

2. Solve for $r$ⓐ when $d=180\,\mathrm{and}\,t=4.5$ⓑ in general

3. Solve for $h$ⓐ when $A=176\,\mathrm{and}\,b=22$ⓑ in general

1. Solve for $b$ⓐ when $A=65\,\mathrm{and}\,h=13$ⓑ in general

2. Solve for the principal, P forⓐ $I=$3,950,r=6\%,$$t=5\,\mathrm{years}\,$ⓑ in general

3. Solve for the time, t forⓐ $I=$624,P=$6,000,$$r=5.2\%$ⓑ in general

1. Solve the formula $5x+2y=10$ for yⓐ when $x=4$ⓑ in general

2. Solve the formula $4x+y=5$ for yⓐ when $x=-3$ⓑ in general

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

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

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

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

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

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

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

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

2. Solve the formula $V=LWH$ for $H$.

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

1. Solve the equation $5x-2y=10$ for $x$ⓐ when $y=10$ⓑ in generalⓒ Which solution is easier for you, ⓐ or ⓑ ? Why?