5.6 Solving a Formula for a Specific Variable

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.

Consider the formula $d=rt$. This formula gives the value of $d$, distance, when you substitute in the values of $r\text{and}t$, the rate and time. But sometimes we have to find the value of $t$. We substitute in values of $d\text{and}r$ and then use 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 $d\text{and}r$. 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. A rearranged quotient is valid only when its denominator is nonzero. If that denominator is zero, check the original equation separately; depending on the other values, every value or no value of the requested variable may satisfy it.

The formula $I=Prt$ 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 the previous examples, 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.

Practice Makes Perfect

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. Round calculated principal amounts in part (a) to the nearest cent. Leave the general formulas in part (b) exact.

In the following exercises, solve.