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!
Before you
...
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!
Before you get started, take this readiness quiz.
1. Solve: 15t = 120.
If you missed this problem, review Example 2.13.
2. Solve: 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 = rt where d = distance
r = rate
t = 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.
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 31
2 hours. What distance has he traveled?
HOW TO : : SOLVE AN APPLICATION (WITH A FORMULA).
Read the problem. Make sure all the words and ideas are understood.
Identify what we are looking for.
Name what we are looking for. Choose a variable to represent that quantity.
Translate into an equation. Write the appropriate formula for the situation. Substitute in the
given information.
Solve the equation using good algebra techniques.
Check the answer in the problem and make sure it makes sense.
Answer the question with a complete sentence.
Step 1.
Step 2.
Step 3.
Step 4.
Step 5.
Step 6.
Step 7.
260 Chapter 2 Solving Linear Equations and Inequalities
This OpenStax book is available for free at http://cnx.org/content/col12116/1.2Solution
Step 1. Read the problem.
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. d = rt
Substitute in the given information. d = 12 · 31 2
Step 5. Solve the equation. d = 42 miles
Step 6. Check
Does 42 miles make sense?
Jamal rides:
Step 7. Answer the question with a complete sentence. Jamal rode 42 miles.
TRY IT : : 2.115
Lindsay drove for 51
2 hours at 60 miles per hour. How much distance did she travel?
TRY IT : : 2.116
Trinh walked for 21
3 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?
Solution
Step 1. Read the problem.
Step 2. Identify what you are looking for. How many hours (time)
Step 3. Name.
Choose a variable to represent it. Let t = time.
Chapter 2 Solving Linear Equations and Inequalities 261Step 4. Translate.
Write the appropriate formula. d = rt
Substitute in the given information. 520 = 65t
Step 5. Solve the equation. t = 8
Step 6. Check. Substitute the numbers into
the formula and make sure the result is a
true statement.
d = rt
520 =? 65 · 8
520 = 520 ✓
Step 7. Answer the question with a complete sentence. Rey’s trip will take 8 hours.
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 = rt . This formula gives the value of d , distance, when you
substitute in the values of r 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 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 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
Solve the formula d = rt for t :
ⓐ when d = 520 and r = 65 ⓑ in general
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.
262 Chapter 2 Solving Linear Equations and Inequalities
This OpenStax book is available for free at http://cnx.org/content/col12116/1.2ⓐ when d = 520 and r = 65 ⓑ in general
Write the formula. d = rt Write the formula. d = rt
Substitute. 520 = 65t
Divide, to isolate t . 520
65 = 65 65t Divide, to isolate t . dr = rt r
Simplify. 8 = t Simplify. dr = t
We say the formula t = dr is solved for t .
TRY IT : : 2.119 Solve the formula d = rt for r :
ⓐ when d = 180 and t = 4 ⓑ in general
TRY IT : : 2.120 Solve the formula d = rt for r :
ⓐ when d = 780 and t = 12 ⓑ in general
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