Literal Equations

Treat the other letters like numbers

In $d = rt$, solving for $r$ means getting $r$ by itself. The $r$ is multiplied by $t$, so divide both sides by $t$: $r = \dfrac{d}{t}$. That's the same move you'd make on $12 = 3r$ — divide by the thing attached to $r$.

Your answer will be an expression, not a number, and that's the point. $r = \dfrac{d}{t}$ is a rebuilt formula: feed it any distance and time and it hands back the rate.

Multi-step literal equations

Formulas with more going on solve in the same order as regular equations: move the terms you don't want first, then divide by whatever is attached to your target. To solve $p = 2l + 2w$ for $w$, first subtract $2l$ from both sides to get $p - 2l = 2w$, then divide by $2$: $w = \dfrac{p - 2l}{2}$.

One care point: when you divide, divide the entire side. $\dfrac{p - 2l}{2}$ keeps both terms over the $2$. Writing $p - \dfrac{2l}{2}$ — dividing only the last term — is the classic error here.

Why bother rearranging?

Because you use formulas in whatever direction the problem needs. $d = rt$ is ready to compute distance, but if a problem gives you distance and time, solving for $r$ once gives you a formula you can reuse forever instead of re-solving the equation with new numbers every time.

This skill is everywhere later: converting between temperature scales, rearranging slope-intercept form, isolating a variable in science class. It's the same three or four moves each time.

Worked examples

Try one yourself

Common questions