0.7 Solving Formulas or Equations for a Specific Variable

Solve a Formula for a Specific Variable

We have all probably worked with some geometric formulas in our study of mathematics. Formulas are used in so many fields, it is important to recognize formulas and be able to manipulate them easily.

It is often helpful to solve a formula for a specific variable. If you need to put a formula in a spreadsheet, it is not unusual to have to solve it for a specific variable first. We isolate that variable on one side of the equals sign with a coefficient of one and all other variables and constants are on the other side of the equal sign.  The properties of equalities from the previous page will be used to isolate specific variables from an equation or formula.

Geometric formulas often need to be solved for another variable, too. The formula [latex]V = \frac{1}{3}πr^{2}h[/latex]  is used to find the volume of a right circular cone when given the radius of the base and height. In the next example, we will solve this formula for the height.


Solve the formula [latex]V = \frac{1}{3}πr^{2}h[/latex] for h.

In the sciences, we often need to change temperature from Fahrenheit to Celsius or vice versa. If you travel in a foreign country, you may want to change the Celsius temperature to the more familiar Fahrenheit temperature.


Solve the formula [latex]C=\frac{5}{9}(F-32)[/latex] for F

We can now use the formula [latex]F=\frac{9}{5}C+32[/latex] to find the Fahrenheit temperature when you are given the Celsius temperature.

The next example uses the formula for the surface area of a right cylinder.


Solve the formula [latex] S = 2πr^2 + 2πrh [/latex] for h.

Sometimes we might be given an equation that is solved for [latex]y[/latex] and needs to be solved for [latex]x[/latex] or vice versa.  In the following example, we are given an equation with both [latex]x[/latex] and [latex]y[/latex] on the same side to be solved for [latex]y[/latex].


Solve the formula 8[latex]x[/latex] + 7[latex]y[/latex] = 15 for [latex]y[/latex]