# Solving Linear Equations and Inequalities

## 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.

### 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  and , we used the formula . This formula gives the value of , distance, when you substitute in the values of , the rate and time. But in , we had to find the value of . We substituted in values of  and then used algebra to solve for . If you had to do this often, you might wonder why there is not a formula that gives the value of  when you substitute in the values of . We can make a formula like this by solving the formula  for .
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.
The formula  is used to calculate simple interest, I, for a principal, P, invested at rate, r, for t years.
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.
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.

### Key Concepts

1. To Solve an Application (with a formula)


2. 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:  where d = distance, r = rate, t = time.
3. 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 other variables and constants on the other side.

### Practice Makes Perfect

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 .
In the following exercises, use the formula .
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.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?