Module 1: Solving Linear Equations with One Variable
1.10 Formulas
section 1.10 Learning Objectives
1.10: Formulas
Evaluate a formula for given values
Solve for a specified variable in a formula
Consider the following problem in the video:
We could have isolated the w in the formula for perimeter before we solved the equation, and if we were going to use the formula many times, it could save a lot of time. The next example shows how to isolate a variable in a formula before substituting known dimensions or values into the formula.
Example 1
Solve for the variable, [latex]w[/latex], in the formula for the perimeter of a rectangle:
You can rewrite the equation so the isolated variable is on the left side.
[latex]w=\frac{p-2l}{2}[/latex]
Rates
There is often a well-known formula or relationship that applies to a word problem. For example, if you were to plan a road trip, you would want to know how long it would take you to reach your destination. [latex]d=rt[/latex] is a well-known relationship that associates distance traveled, the rate at which you travel, and how long the travel takes.
Distance, Rate, and Time
If you know two of the quantities in the relationship [latex]d=rt[/latex], you can easily find the third using methods for solving linear equations. For example, if you know that you will be traveling on a road with a speed limit of [latex]30\frac{\text{ miles }}{\text{ hour }}[/latex] for 2 hours, you can find the distance you would travel by multiplying rate times time or [latex]\left(30\frac{\text{ miles }}{\text{ hour }}\right)\left(2\text{ hours }\right)=60\text{ miles }[/latex].
We can generalize this idea depending on what information we are given and what we are looking for. For example, if we need to find time, we could solve the [latex]d=rt[/latex] equation for t using division:
[latex]d=rt\\\frac{d}{r}=t[/latex]
Likewise, if we want to find rate, we can isolate r using division:
[latex]d=rt\\\frac{d}{t}=r[/latex]
In the following example you will see how this formula is applied to answer a few questions about ultra marathon running.
Ann Trason
Ultra marathon running (defined as anything longer than 26.2 miles) is becoming very popular among women even though it remains a male-dominated niche sport. Ann Trason has broken twenty world records in her career. One such record was the American River 50-mile Endurance Run which begins in Sacramento, California, and ends in Auburn, California.[1] In 1993 Trason finished the run with a time of 6:09:08. The men’s record for the same course was set in 1994 by Tom Johnson who finished the course with a time of 5:33:21.[2]
In the next examples we will use the [latex]d=rt[/latex] formula to answer the following questions about the two runners.
What was each runner’s rate for their record-setting runs?
What was each runner’s time for running one mile?
To make answering the questions easier, we will round the two runners’ times to 6 hours and 5.5 hours.
Example 2
What was each runner’s rate for their record-setting runs?
Show Solution
Read and Understand: We are looking for rate and we know distance and time, so we can use the idea: [latex]d=rt[/latex]
If we divide both sides of the equation with t, we will have equation for the rate:
[latex]\frac{d}{t}=r[/latex]
Define and Translate: Because there are two runners, making a table to organize this information helps. Note how we keep units to help us keep track of what how all the terms are related to each other.
Runner
Distance =
(Rate )
(Time)
Trason
50 miles
r
6 hours
Johnson
50 miles
r
5.5 hours
Write and Solve:
Trason’s rate:
[latex]\frac{d}{t}=r[/latex]
[latex]\begin{array}{c}\frac{50\text{ miles }}{6\text{ hours }}=r\end{array}[/latex]
If we divide 50 with 6 we get the approximate rate in terms of miles per hour (MPH)
Trason’s rate is [latex]\frac{8.33\text{ miles }}{\text{ hour }} = 8.33[/latex] MPH. (rounded to two decimal places)
Johnson’s rate:
[latex]\frac{d}{t}=r[/latex]
[latex]\begin{array}{c}\frac{50\text{ miles }}{5.5\text{ hours }}=r\end{array}[/latex]
If we divide 50 with 5.5 we get the approximate rate in terms of miles per hour (MPH)
Johnson’s rate is [latex]\frac{9.1\text{ miles }}{\text{ hour }} = 9.1[/latex] MPH. (rounded to two decimal places)
Check and Interpret:
We can fill in our table with this information.
Runner
Distance =
(Rate )
(Time)
Trason
50 miles
8.33 [latex]\frac{\text{ miles }}{\text{ hour }}[/latex]
6 hours
Johnson
50 miles
9.1 [latex]\frac{\text{ miles }}{\text{ hour }}[/latex]
5.5 hours
Now we will tackle the second question where we are asked to find a time for each runner.
Example 3
What was each runner’s time for running one mile?
Show Solution
Here is the table we created for reference:
Runner
Distance =
(Rate )
(Time)
Trason
50 miles
8.33 [latex]\frac{\text{ miles }}{\text{ hour }}[/latex]
6 hours
Johnson
50 miles
9.1 [latex]\frac{\text{ miles }}{\text{ hour }}[/latex]
5.5 hours
Read and Understand: We are looking for time, and this time our distance has changed from 50 miles to 1 mile, so we can use [latex]d=rt[/latex].
If we divide both sides of the equation with r, we will have the equation for the time:
[latex]\frac{d}{r}=t[/latex]
Define and Translate: We can use the formula [latex]d=rt[/latex] to make our table. This time the unknown is t, the distance is 1 mile, and we know each runner’s rate. It may help to create a new table:
Runner
Distance =
(Rate )
(Time)
Trason
1 mile
8.33 [latex]\frac{\text{ miles }}{\text{ hour }}[/latex]
t hours
Johnson
1 mile
9.1 [latex]\frac{\text{ miles }}{\text{ hour }}[/latex]
t hours
Write and Solve:
Trason:
Start with the equation from above where we isolated the t.
[latex]\frac{d}{r}=t[/latex]
Now substitute in the values for d=1, and r=8.33 from the table above.
[latex]\frac{1}{8.33}=t[/latex] hours
[latex]0.12\text{ hours }=t[/latex].
0.12 hours is about 7.2 minutes, so Trason’s time for running one mile was about 7.2 minutes. WOW! She did that for 6 hours!
Johnson:
Again, start with:
[latex]\frac{d}{r}=t[/latex]
Now substitute in the values for d=1 and r=9.1 from the table above.
0.11 hours is about 6.6 minutes, so Johnson’s time for running one mile was about 6.6 minutes. WOW! He did that for 5.5 hours!
Check and Interpret:
Have we answered the question? We were asked to find how long it took each runner to run one mile given the rate at which they ran the whole 50-mile course. Yes, we answered our question.
Trason’s mile time was [latex]7.2\frac{\text{minutes}}{\text{mile}}[/latex] and Johnsons’ mile time was [latex]6.6\frac{\text{minutes}}{\text{mile}}[/latex]
In the following video, we show another example of answering many rate questions given distance and time.
Area
The area of a triangle is given by [latex]A=\frac{1}{2}bh[/latex] where
[latex]A[/latex] = area
[latex]b[/latex] = the length of the base
[latex]h[/latex] = the height of the triangle
Remember that when two variables or a number and a variable are sitting next to each other without a mathematical operator between them, you can assume they are being multiplied. This can seem frustrating, but you can think of it like mathematical slang. Over the years, people who use math frequently have just made that shortcut enough that it has been adopted as convention.
In the next example we will use the formula for area of a triangle to find a missing dimension, as well as use substitution to solve for the base of a triangle given the area and height.
Example 4
Find the base, [latex]b[/latex], of a triangle with an area, [latex]A[/latex], of 20 square feet and a height, [latex]h[/latex], of 8 feet.
Show Solution
Use the formula for the area of a triangle, [latex]{A}=\frac{{1}}{{2}}{bh}[/latex].
Substitute the given lengths into the formula and solve for [latex]b[/latex].
We can rewrite the formula in terms of b or h as we did with perimeter previously. This probably seems abstract, but it can help you develop your equation-solving skills, as well as help you get more comfortable with working with all kinds of variables, not just x.
Example 5
Use the multiplication and division properties of equality to isolate the variable [latex]b[/latex]in the Area formula [latex]A=\frac{1}{2}bh[/latex].
Write the equation with the desired variable on the left-hand side as a matter of convention:
[latex]h=\frac{2A}{b}[/latex]
The following video shows another example of finding the base of a triangle given area and height.
Temperature
Let’s look at another formula that includes parentheses and fractions, the formula for converting from the Fahrenheit temperature scale to the Celsius scale.
This formula is telling us that to find the equivalent Celsius temperature, we will start with the Fahrenheit temperature, subtract 32 from it, and then multiply the result by [latex]\frac{5}{9}[/latex]. But what if we are starting with a Celsius temperature and want to convert to Fahrenheit? The example below will show how you can use the same formula to convert from Celsius to Fahrenheit.
Example 6
Given a temperature of [latex]12^{\circ}{C}[/latex], find the equivalent in [latex]{}^{\circ}{F}[/latex].
Show Solution
Substitute the given temperature in[latex]{}^{\circ}{C}[/latex] into the conversion formula:
In the next video, we show another example of converting from celsius to fahrenheit.
As with the other formulas we have worked with, in the examples above, we could have isolated the variable F first, then substituted in the given temperature in Celsius. The example below will show you how to solve the formula for F.
Example 7
The formula below is used for converting from the Fahrenheit scale to the Celsius scale. Solve the formula for [latex]F[/latex]. (This will change it into a formula that can be used to convert Celsius to Fahrenheit).
To isolate the variable [latex]F[/latex], it would be best to clear the fraction involving [latex]F[/latex] first. Multiply both sides of the equation by [latex]\displaystyle \frac{9}{5}[/latex].
Express the formula for the surface area of a cylinder, [latex]s=2\pi rh+2\pi r^{2}[/latex], in terms of the height, [latex]h[/latex].
In this example, the variable [latex]h[/latex] is buried pretty deeply in the formula for surface area of a cylinder. Using the order of operations, it can be isolated. Before you look at the solution, use the box below to write down what you think is the best first step to take to isolate [latex]h[/latex].
Show Solution
Isolate the term containing the variable, h, by subtracting [latex]2\pi r^{2}[/latex]from both sides.
