Solving For a Specific Variable

Learning Objective(s)

·         Rearrange formulas to isolate specific variables.

Introduction

Many real world events have well-known equations that describe relationships among different quantities. These familiar, frequently used equations are called formulas. You have probably used formulas to figure out things like the area of a rectangle (area = width • height), the speed of a moving object (speed = distance/time), or the length of one side of a triangle ( ).  Most mathematical formulas include more than one variable. Although they have a special name, formulas are written and solved like any other equations.

Rearranging Formulas to Solve for a Different Variable

Formulas are written so that one variable is already isolated. All you have to do to solve for that variable is to substitute numbers for the variables and calculate the results. For instance, you can figure out the volume of a rectangle using the formula (volume = length • width • height), just by plugging in the values of l, w, and h. It’s easy.

Another great and useful thing about formulas is that we can rearrange them to make them more useful under different circumstances. We can group the variables we know on one side and the one we want to find on the other. If we already knew the volume, height, and width of a rectangle, we could rewrite the formula so that it solves for the length instead of the volume.

We rearrange formulas the same way we solve equations, using tools like the Distributive Property, Inverse Operations, multiplying by a common denominator, and the Properties of Equality to move terms around. It may seem like a lot of trouble to go through, but it is worth it if we need to use a formula repeatedly to find a variable other than the one it was originally written for.

Rewrite the formula for the volume of a rectangle, , to solve for width ( ).

A) B) C) D) Rearranging Multi-step Formulas to Solve for a Different Variable

Some formulas can be written to solve for a different variable with just a single step. Many others are multi-step formulas and these will require a series of actions to reorganize. No need to be intimidated, though. It can be done one step at a time.

For example, let’s look at the formula for the perimeter of a rectangle, . It’s helpful to know that the perimeter equals 2 times the length plus 2 times the width, if you know the length and width. But what if for some odd reason you are given the perimeters and lengths of different rectangles, and are asked to calculate their widths? You would rewrite the formula so that it solves for w.

We’ll take it one step at a time. First, we’ll isolate the term containing the variable we’re interested in, 2w, by subtracting the other term in its expression, 2l. Of course, we’ll do that to both sides. That gives us .

Next, we can clear the co-efficient of w by multiplying by its reciprocal, . Distribute the multiplication, and the end result is . Now we have a formula that solves for width (w).      Summary

Formulas are a type of equation, one that usually contains multiple variables and describes important relationships and frequently made calculations. Although they are written to solve for a particular variable, formulas can be rearranged following standard algebraic rules so that they instead solve for one of the other variables.