Changing the Subject of a Formula

Why this matters

A formula is usually written to find one particular thing. $A = \pi r^2$ gives the area from the radius — but a gardener who wants a circular lawn of 50 m² needs the radius from the area. Rearranging lets one formula answer every question about the same relationship.

Method 1: function machines

Draw the machine that builds the formula, starting from the letter you want as the subject. For $V = \frac{1}{3}\pi r^2 h$ starting from $r$:

$r \rightarrow \boxed{\text{square}} \rightarrow \boxed{\times \pi h} \rightarrow \boxed{\div 3} \rightarrow V$

Run it in reverse with inverse operations:

$V \rightarrow \boxed{\times 3} \rightarrow \boxed{\div \pi h} \rightarrow \boxed{\sqrt{\ }} \rightarrow r \qquad \text{so} \qquad r = \sqrt{\frac{3V}{\pi h}}$

Function machines are quick, but they only work when the letter appears once.

Method 2: the balance method

A formula is like an equation — it stays balanced if you do the same thing to both sides. This method always works, and it is the one to use when things get awkward.

Two things that catch people out

The letter is negative. In $M = 4(t - x)$ you want $+x$, not $-x$. Add $x$ to both sides early, and the rest is straightforward.

Dividing by an expression. When you divide by $(g - y)$, keep it in brackets while you work. Once it sits under a long fraction line, the line acts as a bracket and you can drop them: $x = \frac{f}{g - y}$.

Fractions in front. To remove $\frac{2}{5}$ in $P + e = \frac{2}{5}x$, multiply both sides by $\frac{5}{2}$: $x = \frac{5(P + e)}{2}$.