Though mathematical, formulas are the backbone of understanding content from many areas of study. They are useful in the sciences and social sciences—fields such as chemistry, physics, biology, psychology, sociology, and criminal justice. Healthcare workers use formulas, too, even for something as routine as dispensing medicine. The widely used spreadsheet program Microsoft Excel^TM relies on formulas to do its calculations. Many teachers use spreadsheets to apply formulas to compute student grades. It is important to be familiar with formulas and be able to manipulate them easily.

A common formula is [latex]d=rt[/latex] for calculating distance based on rate and time. This formula gives the value of [latex]d[/latex] when you substitute in the values of [latex]r[/latex] and [latex]t[/latex]. But what if you have to find the value of [latex]t[/latex]. We would need to substitute in values of [latex]d[/latex] and [latex]r[/latex] and then use algebra to solve for [latex]t[/latex]. If you had to do this often, you might wonder why there isn’t 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 get 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 get that variable by itself with a coefficient of [latex]1[/latex] on one side of the equation and all the other variables and constants on the other side. We will call this solving an equation for a specific variable in general. This process is also called solving a literal equation. The result is another formula, made up only of variables. The formula contains letters, or literals.

Let’s try a few examples, starting with the distance, rate, and time formula we used above.

We can use the formula [latex]A=\Large\frac{1}{2}\normalsize bh[/latex] to find the area of a triangle when we are given the base and height. In the next example, we will solve this formula for the height.

Previously, we used the formula [latex]I=Prt[/latex] to calculate simple interest, where [latex]I[/latex] is interest, [latex]P[/latex] is principal, [latex]r[/latex] is rate as a decimal, and [latex]t[/latex] is time in years.

Watch the following video to see another example of how to solve an equation for a specific variable.

Later in this class, and in 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 you 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]. To do this, we will follow the same steps that we used to solve a formula for a specific variable.

In the previous examples, we used the numbers in part (a) as a guide to solving algebraically in part (b). Do you think you’re ready to solve a formula in general without using numbers as a guide?

In the following video we show another example of how to solve an equation for a specific variable.