Solve for a Variable

A variable is a letter standing for a number you do not yet know. "Solve for the variable" means rewrite the equation until that letter sits alone on one side and everything else sits on the other.

Two situations look different but obey the same rules:

• Numeric equations such as $5x - 7 = 3x + 9$, where the answer is a number.
• Literal equations (formulas) such as $A = \tfrac{1}{2}bh$, where the answer is another expression: $h = \dfrac{2A}{b}$.

The second kind is the one students find alien, but nothing changes. The letters $A$ and $b$ are simply numbers whose values have not been announced. Rearranging a formula once and substituting afterwards beats substituting first and rearranging every single time.

The skill transfers directly into science and finance, where almost every formula gets used in more than one direction. The same relationship between distance, speed and time is three different formulas depending on which quantity is missing, and rearranging is what turns one memorised statement into all three.

Whatever you do to one side, do to the other. That single rule keeps the equation balanced.

The procedure

1. Clear brackets and fractions. Distribute; multiply through by the common denominator.
2. Gather the target variable on one side. Add or subtract terms so every copy of the variable is together, usually on the side where its coefficient is larger.
3. Undo addition and subtraction attached to it.
4. Undo multiplication and division last, dividing by the coefficient.
5. Substitute back into the original equation to verify.

The order matters: you unwrap an expression in the opposite order to how it was built, the way you take off a coat before a sweater.

• Dividing before collecting. Divide by $2$ while a $+7$ is still glued on and you must divide the $7$ too. Undo the addition first.
• Multiplying only one term by the denominator. Clearing a fraction means multiplying every term on both sides.
• Panicking at letters. In $A = \tfrac{1}{2}bh$, treat $A$ and $b$ exactly as you would $24$ and $6$.
• Losing the negative coefficient. From $-x = 5$ the answer is $x = -5$, not $x = 5$.
• Skipping the check. Substituting your value back into the original equation catches nearly every sign slip in under ten seconds.