Rearranging FormulasCambridge IGCSE Maths: Revision notes

The subject of a formula is the letter on its own on one side of the equals sign (e.g. $A$ is the subject of $A = \pi r^2$). Changing the subject means rearranging the formula so a different letter becomes isolated on one side instead.

The golden rule: whatever operation you perform, do it to both sides of the formula, in the same way you would solve an equation.

When the required subject appears only once and has no power or root attached, use inverse operations step by step, undoing each operation in reverse order (like unpicking BIDMAS backwards).

Example: make $x$ the subject of $y = 3x + 4$

1. Subtract 4 from both sides: $y - 4 = 3x$
2. Divide both sides by 3: $x = \frac{y-4}{3}$

If the subject has a power, use the inverse root to isolate it; if it has a root, square both sides.

Example: make $r$ the subject of $A = \pi r^2$

1. Divide both sides by $\pi$: $\frac{A}{\pi} = r^2$
2. Square root both sides: $r = \sqrt{\frac{A}{\pi}}$

Example: make $x$ the subject of $y = \sqrt{x} + 3$

1. Subtract 3: $y - 3 = \sqrt{x}$
2. Square both sides: $x = (y-3)^2$

When the required subject appears in more than one term, collect all terms containing it on one side, factorise it out, then divide.

Example: make $x$ the subject of $ax + b = cx + d$

1. Collect $x$-terms on one side: $ax - cx = d - b$
2. Factorise: $x(a-c) = d - b$
3. Divide: $x = \frac{d-b}{a-c}$