Changing the subject of a formula, including formulae involving brackets, fractions, squares and square roots.

What this dot point is asking

The SQA wants you to rearrange a formula so that a different letter becomes the subject (the variable on its own, usually on the left). This includes formulae with brackets, fractions, squares and square roots, and the skill is the same balancing process used to solve equations.

The balancing method

A formula is just an equation with letters, so the rules of solving equations apply. To isolate a chosen variable, peel away everything else by doing the inverse operation to both sides, working from the outside in.

The order matters: undo the operations in the reverse order to how they were applied. Here $x$ was multiplied by $3$ then had $4$ added, so you remove the $+4$ first, then divide by $3$.

Formulae with fractions and brackets

When the subject is trapped in a fraction, multiply both sides by the denominator to free it. When it is inside a bracket, you can expand first or divide by the multiplier outside.

Formulae with squares and roots

A square is undone by a square root, and a square root is undone by squaring. These come up in area, Pythagoras and volume formulae.

Formulae with the subject in two places

A harder case has the chosen variable appearing more than once. Gather all the terms containing it on one side, factorise it out of a bracket, then divide by the bracket.

The factorising step is the key: whenever the subject appears twice, collect it and pull it out of a bracket before the final division.

A word on the order of moves

Choosing the order well keeps the algebra clean. Deal with whatever is furthest from the subject first: remove added or subtracted terms, then multipliers and divisors, then powers or roots. Working from the outside in mirrors how you would undo the operations on a number. A helpful check is to substitute a simple number for the original subject, work out the formula, then confirm your rearranged version gives that number back; if it does, the rearrangement is sound.

Examples in context

Rearranging formulae lets you answer the question actually asked. The formula for the circumference of a circle is $C = 2\pi r$; if you know the circumference of a tree trunk and want its radius, make $r$ the subject: $r = \dfrac{C}{2\pi}$. Scientists rearrange constantly, for instance turning $v = u + at$ into $t = \dfrac{v - u}{a}$ to find how long an acceleration takes. The same formula serves many questions once you can rearrange it.

Try this

Q1. Make $x$ the subject of $y = 5x - 2$. [2 marks]

• Cue. $x = \dfrac{y + 2}{5}$.

Q2. Make $b$ the subject of $A = \dfrac{1}{2}bh$. [2 marks]

• Cue. $b = \dfrac{2A}{h}$.

Q3. Make $x$ the subject of $y = x^2 + 3$ (for $x > 0$). [2 marks]

• Cue. $x = \sqrt{y - 3}$.