Changing the Subject of a Formula Revision Notes | GCSE AQA Higher Maths

1. Why Changing the Subject Matters

You already know how to solve equations — you isolate the unknown by doing the same thing to both sides. Changing the subject of a formula is exactly the same skill, applied to letters instead of numbers.

Think about the formula for speed: $v = u + at$. If you know $v$, $u$, and $t$ but need to find $a$, you need $a$ as the subject. Rearranging gives you a new formula you can use directly — no guesswork needed.

This skill appears throughout GCSE Maths and Science. Mastering it means you can work with any formula, regardless of which variable you are asked to find.

2. The Core Method: Inverse Operations

The golden rule is: whatever has been done to the variable you want, undo it — in reverse order. Think of it like unwrapping a parcel: the last thing wrapped is the first thing you remove.

Here is the order of inverse operations to keep in mind. If the target variable has been added to something, subtract. If it has been subtracted, add. If it has been multiplied, divide. If it has been divided, multiply. If it has been squared, take the square root. If it is inside a square root, square both sides. Always work from the outside in and undo the last operation first.

Notice how each step undoes one operation. Subtracting $u$ removes the addition; dividing by $t$ removes the multiplication. The variable $a$ is now the subject.

3. Worked Examples: Subject Appears Once

When the target variable appears exactly once in the formula, the method is a clean sequence of inverse operations. Work through the following examples carefully — each one introduces a slightly different structure.

Worked example: Rearranging a Linear Formula

Make $u$ the subject of $v = u + at$.

1. $u$ has $at$ added to it, then the result equals $v$. — Understanding the structure tells us what to undo.
2. $v - at = u + at - at$ — Subtraction undoes addition. [M1]
3. $v - at = u$ — The $at$ terms cancel on the right.
4. $u = v - at$ — Convention is to write the subject on the left. [A1]

Answer: $u = v - at$

Worked example: Rearranging a Formula Involving a Square

Make $r$ the subject of $A = \pi r^2$.

1. $\frac{A}{\pi} = r^2$ — Division undoes multiplication by $\pi$. [M1]
2. $\sqrt{\frac{A}{\pi}} = r$ — Square-rooting undoes squaring. [M1]
3. $r = \sqrt{\frac{A}{\pi}}$ — At GCSE, take the positive root unless told otherwise. [A1]

Answer: $r = \sqrt{\dfrac{A}{\pi}}$

Worked example: Rearranging a Formula with a Square Root

Make $l$ the subject of $T = 2\pi\sqrt{\dfrac{l}{g}}$.

1. $\frac{T}{2\pi} = \sqrt{\frac{l}{g}}$ — Removes the multiplication by $2\pi$ on the right. [M1]
2. $\left(\frac{T}{2\pi}\right)^2 = \frac{l}{g}$ — Squaring undoes the square root. [M1]
3. $g\left(\frac{T}{2\pi}\right)^2 = l$ — Multiplication undoes division by $g$. [M1]
4. $l = \frac{gT^2}{4\pi^2}$ — Expanding $\left(\frac{T}{2\pi}\right)^2 = \frac{T^2}{4\pi^2}$, then multiply by $g$. [A1]

Answer: $l = \dfrac{gT^2}{4\pi^2}$

4. When the Subject Appears More Than Once

Sometimes the variable you want appears in two or more separate terms. You cannot isolate it with a single inverse operation — you need to collect all those terms on one side first, then factorise.

This is where many students get stuck. The key insight is: once you have factorised, the variable you want appears only once — as a factor outside the bracket — and you can then divide to isolate it.

Worked example: Rearranging When the Subject Appears Twice

Make $x$ the subject of $ax + b = cx + d$.

1. $ax - cx = d - b$ — Subtract $cx$ from both sides and subtract $b$ from both sides. [M1]
2. $x(a - c) = d - b$ — $x$ is a common factor of $ax$ and $cx$. Factorising creates a single term containing $x$. [M1]
3. $x = \frac{d - b}{a - c}$ — Division isolates $x$. This only works when $a eq c$. [A1]

Answer: $x = \dfrac{d - b}{a - c}$

Worked example: Rearranging with the Subject in a Denominator

Make $t$ the subject of $\dfrac{a}{t} + b = c$.

1. $\frac{a}{t} = c - b$ — Removes the $+b$ term. [M1]
2. $a = t(c - b)$ — Clears $t$ from the denominator. [M1]
3. $t = \frac{a}{c - b}$ — Isolates $t$. Valid when $c eq b$. [A1]

Answer: $t = \dfrac{a}{c - b}$

5. Formula Reference and Practice

Here is a compact summary of the rearrangement moves you need for GCSE. Each row shows the structure, the inverse operation, and the result — use this as a quick reference when working through problems.

6. Exam Technique and Common Mistakes

Changing the subject is a reliable source of method marks on both calculator-allowed questions and questions without a calculator. The steps are always the same — but there are predictable places where marks get dropped.

In questions without a calculator, you must show every algebraic step clearly. On calculator-allowed exam questions , the method is identical — the calculator only helps if you are substituting numbers, not rearranging letters.

That final example brings together every technique from this lesson: clearing a fraction, dividing by a product of constants, and applying a root to undo a power. If you can work through that confidently, you have the full toolkit for changing the subject at GCSE.

Changing the subject of a formula means isolating a chosen variable by applying inverse operations to both sides in reverse order. When the subject appears once, work through the operations systematically. When it appears more than once, collect, factorise, then divide.

7. Lesson Summary

• Apply inverse operations in reverse order: undo the outermost operation first.
• To remove a square root, square both sides; to remove a square, take the square root.
• To clear a fraction, multiply both sides by the denominator before doing anything else.
• When the subject appears in two or more terms: collect on one side, factorise, then divide.
• Always check by substituting numbers into both the original and rearranged formulae.

Every rearrangement question on the GCSE Higher paper rewards method marks — even a partially correct rearrangement can earn marks if your steps are clearly shown. Write each step on a new line and never skip an operation.