Literal Equations and Formulas

A literal equation is simply an equation that contains two or more variables. You work with literal equations all the time in math, science, and geometry whenever you use a formula.

Sometimes, a formula isn't set up the way you need it. For example, if you know the distance ($d$) and the rate ($r$), but want to find the time ($t$), the standard formula $d = rt$ isn't as helpful as it could be. You need to rearrange the formula to solve for the specific variable you want.

The Golden Rule of Rearranging

When solving a literal equation for a specific variable, treat the target variable as your unknown (like you normally treat $x$), and treat all the other variables as if they were regular numbers.

Use the exact same inverse operations you use to solve basic algebraic equations:

• If a variable is added, subtract it from both sides.
• If a variable is multiplied, divide both sides by it.

Example 1: Area of a Triangle

Problem: Solve the area formula $A = \frac{1}{2}bh$ for $h$.

In this problem, we want to isolate $h$. We need to move the $\frac{1}{2}$ and the $b$ to the other side of the equation.

1. Clear the fraction: Multiply both sides by $2$. $2 \cdot A = 2 \cdot \left(\frac{1}{2}bh\right)$ $2A = bh$

2. Isolate $h$: Since $b$ is multiplied by $h$, divide both sides by $b$. $\frac{2A}{b} = \frac{bh}{b}$ $h = \frac{2A}{b}$

Now you have a brand new formula that calculates the height ($h$) directly!

Example 2: Perimeter of a Rectangle

Problem: Solve the perimeter formula $P = 2l + 2w$ for $w$.

Here, $w$ is our target. Treat $P$ and $l$ just like regular numbers.

1. Move the $l$ term away from the $w$ term: Subtract $2l$ from both sides. $P - 2l = 2w$

2. Isolate $w$: Since $w$ is multiplied by $2$, divide the entire other side by $2$. $\frac{P - 2l}{2} = w$

You can also write this as $w = \frac{P}{2} - l$. Both answers are perfectly correct.

Quick Tips for Success

• Highlight the target: If you get confused, circle or lightly highlight the variable you are trying to solve for so you don't lose track of it.
• Reverse PEMDAS: Just like solving regular equations with numbers, undo addition and subtraction first, then undo multiplication and division.