A formula like $P = 2l + 2w$ is an equation with several letters. Sometimes you know the perimeter and the width and want the length, so you solve for $l$. The moves are the same ones that keep an equation balanced; you just treat the other letters as numbers. Knowing the terms and factors of each side tells you what to undo.
| With numbers | With letters |
|---|---|
| $3x + 5 = 20$ | $ax + b = c$ |
| $3x = 20 - 5$ | $ax = c - b$ |
| $x = \frac{20 - 5}{3}$ | $x = \frac{c - b}{a}$ |
Por qué funciona
Each letter stands for a number, so the properties of equality work for letters exactly as they do for numbers. The only new care is division: dividing by a letter is allowed only when that letter is not $0$. And when you divide a side, you divide the whole side, which is why the result is a fraction with the whole side on top.
Idea clave
Find the letter you want, notice what is being done to it, and undo those operations in reverse order, doing the same thing to both sides each time.
Worked examples
Ejemplo resuelto: Perimeter
Solve $P = 2l + 2w$ for $l$.
- $l$ is multiplied by $2$, then $2w$ is added. Undo the addition first: subtract $2w$ from both sides: $P - 2w = 2l$.
- Undo the multiplication: divide both sides by $2$: $\frac{P - 2w}{2} = l$.
Respuesta: $l = \frac{P - 2w}{2}$ (also $l = \frac{P}{2} - w$).
Ejemplo resuelto: Slope-intercept form
Solve $y = mx + b$ for $x$.
- Subtract $b$ from both sides: $y - b = mx$.
- Divide both sides by $m$ (assume $m \ne 0$): $x = \frac{y - b}{m}$. The whole side $y - b$ is divided.
Respuesta: $x = \frac{y - b}{m}$
Ejemplo resuelto: The letter appears twice
Solve $A = P + Prt$ for $P$.
- $P$ is in two terms, so factor it out: $A = P(1 + rt)$.
- Now $P$ is multiplied by $(1 + rt)$. Divide both sides by it: $P = \frac{A}{1 + rt}$.
Respuesta: $P = \frac{A}{1 + rt}$
Error común
Dividing only one term, writing $x = y - \frac{b}{m}$ instead of $x = \frac{y - b}{m}$. Check with numbers: if $y = 11$, $b = 3$ and $m = 2$, then $11 = 2x + 3$ gives $x = 4$. The correct form gives $\frac{11 - 3}{2} = 4$; the slip gives $11 - 1.5 = 9.5$.