Solving for a variable

Start with numbers

A rectangle has an area of $24$ and a length of $6$. How wide is it?

You probably thought "$24 \div 6 = 4$" without writing any equation. Notice what you did: you divided the area by the length. That move works for any rectangle, whatever its numbers.

Formulas are equations too

A formula like $A = lw$ says the same thing for every rectangle at once. Here it is with letters in place of numbers:

Solving for a variable means rearranging a formula so one variable is alone on one side. It uses the same moves as before. You're just undoing operations on letters instead of numbers. Since $w$ is multiplied by $l$, divide both sides by $l$, exactly as you divided $24$ by $6$.

Now it answers every rectangle question at once: if $A = 35$ and $l = 7$, then $w = \frac{35}{7} = 5$.

The distance formula $d = rt$ works the same way. Driving $120$ miles at $40$ miles per hour takes $120 \div 40 = 3$ hours, so $t = \frac{d}{r}$.

Two steps

The perimeter of a rectangle is $P = 2l + 2w$: two lengths and two widths around the edge. To find the length when you know $P$ and $w$, think it through with numbers first. If $P = 20$ and $w = 3$, the two widths use up $6$, leaving $14$ for the two lengths, so each length is $7$.

The letters follow the same path: subtract the widths, then split what's left in two.

A familiar example

Converting Celsius to Fahrenheit uses $F = \frac{9}{5}C + 32$. Solve for $C$ to convert the other way.

Multiplying by $\frac{5}{9}$ undoes multiplying by $\frac{9}{5}$. Check it with a temperature you know: water boils at $212°F$, and $\frac{5}{9}(212 - 32) = \frac{5}{9}(180) = 100°C$ ✓

Key ideas

• A formula is an equation with several variables.
• To solve for one variable, undo the operations around it, doing the same to both sides.
• Treat the other letters like numbers, and test with easy numbers when unsure.