One formula you will use often in algebra and everyday life is the distance formula travelled by an object moving at a constant rate. Rate is an equivalent word for “speed.” The basic idea of rate may already be familiar to you. Do you know what distance you travel if you drive at a steady rate of [latex]60[/latex] miles per hour for [latex]2[/latex] hours? (This might happen if you use your car’s cruise control while driving on the highway.) If you said [latex]120[/latex] miles, you already know how to use this formula!

For an object moving at a uniform (constant) rate, the distance travelled, the elapsed time, and the rate are related by the 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 1

Jamal rides his bike at a uniform rate of [latex]12[/latex] miles per hour for [latex]3\frac{1}{2}[/latex] hours. What distance has he travelled?

Solution

Step 1: Read the problem.

Step 2: Identify what you are looking for.

distance travelled

Step 3: Name. Choose a variable to represent it.

Let [latex]d[/latex] = distance.

Step 4: Translate: Write the appropriate formula.

[latex]\begin{align*} d&=rt\\ d&=?\\ r&=12\;mph\\ t&=3\frac12\;hours \end{align*}[/latex]

Step 5: Substitute in the given information.

[latex]d=12\cdot3\frac{1}{2}[/latex]

Step 6: Solve the equation.

[latex]d=42\;miles[/latex]

Step 7: Check
Does 42 miles make sense?

Jamal rides:

Step 8: Answer the question with a complete sentence.

Jamal rode 42 miles.

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 3.6.1 and Example 3.6.2, we used the formula [latex]d = rt[/latex]. This formula gives the value of [latex]d[/latex], distance, when you substitute in the values of [latex]r[/latex] and [latex]t[/latex], the rate and time. But in Example 3.6.2 we had to find the value of [latex]t[/latex]. We substituted in values of [latex]d[/latex] and [latex]r[/latex] and then used algebra to solve for [latex]t[/latex]. If you had to do this often, you might wonder why there is not a formula that gives the value of [latex]t[/latex] when you substitute in the values of [latex]d[/latex] and [latex]r[/latex]. We can make a formula like this by solving the formula [latex]d=rt[/latex] for [latex]t[/latex].

To solve a formula for a specific variable means to isolate that variable on one side of the equals sign with a coefficient of [latex]1[/latex]. 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.

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