How to Rearrange Formulae (Change the Subject): KS3 Maths

To change the subject of a formula, apply inverse operations to both sides to isolate the new subject — exactly like solving an equation, but with letters instead of numbers. Whatever you do to one side, do to the other. Work outwards from the subject letter, undoing each operation in reverse order.

What does "change the subject" mean?

A formula has a subject — the letter on its own, usually on the left. Changing the subject means rearranging so a different letter is isolated. For example, the formula for speed is s = d ÷ t. Rearranging to make d the subject gives d = s × t; rearranging to make t the subject gives t = d ÷ s. The formula describes the same relationship every time — you are just choosing which variable to solve for.

What is the inverse-operations approach?

Work through the operations in reverse order, applying the inverse of each one to both sides. A useful trick is to imagine substituting a number for the new subject and writing down every operation applied to it — then undo them in reverse.

Operation done to the subject | Inverse to apply
Added (+ a) | Subtract a
Subtracted (− a) | Add a
Multiplied (× a) | Divide ÷ a
Divided (÷ a) | Multiply × a
Squared (²) | Square root (√)

Worked example: one-step rearrangement

Make u the subject of v = u + at.

u has at added to it. Subtract at from both sides.

v − at = u

Write with the new subject on the left: u = v − at

Worked example: two-step rearrangement

Make r the subject of P = 2r + 6.

First, r is multiplied by 2, then 6 is added. Undo in reverse: first subtract 6 from both sides.

P − 6 = 2r

Now divide both sides by 2.

r = (P − 6) ÷ 2

Worked example: subject inside a fraction

Make m the subject of F = ma ÷ 2.

m is multiplied by a, then divided by 2. Undo in reverse: first multiply both sides by 2.

2F = ma

Divide both sides by a.

m = 2F ÷ a

What if the subject appears more than once?

This is a GCSE-level extension, but it appears in harder KS3 questions too. Collect all terms containing the new subject on one side, factorise, then divide. For example, to make x the subject of ax + b = cx + d:

Subtract cx from both sides: ax − cx + b = d

Subtract b: ax − cx = d − b

Factorise: x(a − c) = d − b

Divide: x = (d − b) ÷ (a − c)

How do you check a rearrangement is correct?

Substitute simple numbers into both the original and your rearranged formula and check they give the same result. For P = 2r + 6 rearranged to r = (P − 6) ÷ 2: let r = 5, so P = 2(5) + 6 = 16. Check: (16 − 6) ÷ 2 = 5. ✓

Why is this skill important?

Rearranging formulae is used constantly in science — for example, isolating m from F = ma, or R from V = IR. Being fluent at KS3 saves time in every physics and chemistry lesson, and the same skill carries directly into GCSE algebra.

Frequently asked questions

What does it mean to make a letter the subject of a formula?

It means rearranging the formula so that letter is alone on one side (usually the left), with everything else on the other side. For example, making t the subject of v = u + at gives t = (v − u) ÷ a.

What is the order of operations when rearranging?

Undo operations in the reverse of the order they were applied to the subject letter. If the subject was first multiplied then had something added, first subtract (undo the addition), then divide (undo the multiplication).

What if there is a square root or power?

Apply the inverse: square both sides to undo a square root, or take the square root of both sides to undo a square. For example, to make r the subject of A = πr²: divide by π to get r² = A ÷ π, then square root both sides to get r = √(A ÷ π).

How do I avoid sign errors when rearranging?

Keep the sign attached to the term, not to the operation. When you move −3 from one side, you are adding 3 to both sides, so the −3 becomes +3 on the other side. Write each step on a new line and check the signs after each move.

For Socratic algebra practice at KS3 and GCSE, see aitutors.me.