Literal Equations: Changing the Subject of a Formula

Rearranging a formula to make a different letter the subject — the same inverse operations as solving an equation, but the answer is an expression instead of a number.

"Make r the subject" simply means: rearrange until r stands alone on the left. The method is identical to solving an equation — the only difference is that your answer contains letters instead of a number.

💡

Treat every other letter as if it were an ordinary number. If you know how to solve 3x + 5 = 20, you can rearrange ax + b = c.

1One-step and two-step rearrangements

Worked ExampleMake x the subject: y = 3x + 4

1
Subtract 4 from both sides: y − 4 = 3x.
Undo the +4 first.
2
Divide both sides by 3.
Undo the ×3.
3
x = y − 43
The whole numerator is divided by 3 — keep it in one fraction.

⚠️

Writing x = y − 43 is wrong. The entire left side was divided by 3, so the whole of y − 4 sits on top of the fraction.

2When the subject is inside a product

Worked ExampleMake b the subject: A = 12bh

1
Multiply both sides by 2: 2A = bh.
Clear the fraction first.
2
Divide both sides by h.
3
b = 2Ah

3When the subject is squared or under a root

Worked ExampleMake r the subject: A = πr²

1
Divide both sides by π: Aπ = r².
2
Take the square root of both sides.
The inverse of squaring.
3
r = Aπ
A radius is a length, so only the positive root is used.

4When the subject appears twice

Worked ExampleMake x the subject: ax = bx + c

1
Gather the x terms on one side: ax − bx = c.
Move bx across.
2
Factorise out x: x(a − b) = c.
This is the key step — x must appear once only.
3
x = ca − b
Divide by the whole bracket.

💡

If the letter you want appears more than once, collect its terms and factorise it out. There is no other route.