Rearranging Formulae: How to Change the Subject of a Formula

How to make any letter the subject of a formula, from $v = u + at$ to $A = \pi r^2$ , including formulae where the subject appears twice or sits under a root.

3 min read Supports 3 lessons

A formula like $v = u + at$ tells you $v$ if you know $u$ , $a$ and $t$ . But what if you know $v$ , $u$ and $a$ , and want $t$ ? Rather than solving from scratch every time, rearrange once so that $t$ is the subject.

Key idea

Treat every other letter as if it were a number. Then do exactly what you would do to solve an equation.

A two-step rearrangement

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

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Tip

Compare with solving $10 = 4 + 2t$ : subtract $4$ , divide by $2$ . The steps are identical; only the numbers have become letters.

Powers and roots

To undo a square, take a square root of both sides. To undo a square root, square both sides.

Worked example

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

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Divide by $\pi$ .
$\frac{A}{\pi} = r^2$

Here we keep only the positive root because a radius cannot be negative. In general, $x^2 = k$ gives $x = \pm\sqrt{k}$ .

When the subject appears twice

If the letter you want appears in more than one term, gather those terms on one side and factorise it out.

$ax + b = cx + d$
Make $x$ the subject. It appears twice.

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Fractions

If the subject is in a denominator, multiply both sides by that denominator first to bring it up.

Worked example

Make $R$ the subject of $I = \dfrac{V}{R}$ .

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Multiply both sides by $R$ .
$IR = V$

Watch out

Do every operation to the whole of each side. From $v = u + at$ , dividing only $at$ by $a$ would give the wrong answer.

A link to inverse functions

Rearranging $y = 3x - 6$ to $x = \frac{y + 6}{3}$ is exactly how you find an inverse function: undo each operation in reverse order.

Common questions

What does it mean to change the subject of a formula?: It means rearranging the formula so a different letter is on its own on one side. For example $v = u + at$ rearranged for $t$ gives $t = \frac{v - u}{a}$ .
Is rearranging a formula the same as solving an equation?: Yes, the steps are the same: apply inverse operations to both sides. The only difference is that the other quantities are letters instead of numbers.

Practise rearranging formulae

The Rearranging Formulas lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

What does it mean to change the subject of a formula?

It means rearranging the formula so a different letter is on its own on one side. For example $v = u + at$ rearranged for $t$ gives $t = \frac{v - u}{a}$ .

Is rearranging a formula the same as solving an equation?

Yes, the steps are the same: apply inverse operations to both sides. The only difference is that the other quantities are letters instead of numbers.