What you'll learn
Rearranging formulae (changing the subject) means rewriting a formula to make a different variable the subject. In this guide you will learn the inverse operations used to rearrange, how to change the subject step by step, how to handle powers, roots and brackets, and how to deal with the subject appearing more than once. This is an essential algebra skill for formulae across maths and science.
Key terms and definitions
Subject — the variable on its own, usually on the left of the equals sign.
Change the subject — rearrange so a different variable is alone.
Inverse operation — the operation that undoes another (e.g. + and −, × and ÷).
Formula — a relationship between variables.
Balance — keeping both sides equal by doing the same to each.
Core concepts
Using inverse operations
To change the subject, undo the operations around the variable you want, using inverse operations, while keeping the formula balanced (do the same to both sides). The inverses are: + and −, × and ÷, powers and roots. Work in the reverse order of BIDMAS — undo addition/subtraction before multiplication/division, and so on.
Step-by-step rearranging
Isolate the desired variable one step at a time. For example, to make x the subject of y = 3x + 2:
- Subtract 2: y − 2 = 3x.
- Divide by 3: (y − 2)/3 = x.
So x = (y − 2)/3.
Powers and roots
If the variable is squared, take the square root to undo it; if it is under a square root, square both sides. For example, to make r the subject of A = πr²: divide by π (A/π = r²), then square root (r = √(A/π)).
Brackets
If the variable is inside brackets, you can either expand first or divide by the bracket's multiplier. For example, y = a(x + b): divide by a (y/a = x + b), then subtract b (x = y/a − b).
Subject appearing twice
If the new subject appears more than once, collect those terms on one side, factorise out the variable, then divide. For example, to make x the subject of ax = bx + c: ax − bx = c → x(a − b) = c → x = c/(a − b).
Worked examples
Example 1: Simple rearrange
Make x the subject of y = 5x − 7.
Add 7: y + 7 = 5x. Divide by 5: x = (y + 7)/5.
Example 2: With a power
Make r the subject of V = r³.
Cube root both sides: r = ∛V.
Example 3: Subject appears twice
Make x the subject of 4x + 3 = 2x + k.
4x − 2x = k − 3 → 2x = k − 3 → x = (k − 3)/2.
Common mistakes and how to avoid them
Doing operations in the wrong order. Undo in reverse BIDMAS order.
Not balancing both sides. Whatever you do to one side, do to the other.
Forgetting to root or square fully. A squared variable needs the square root of the whole other side.
Not collecting a repeated subject. Gather those terms, factorise, then divide.
Sign errors. Track signs carefully when moving terms.
Exam technique for Rearranging Formulae
Identify the new subject and the operations around it.
Undo step by step using inverse operations, in reverse BIDMAS order.
Keep the formula balanced at every step.
Use roots/powers to undo squares and square roots.
Factorise when the subject appears more than once.
Quick revision summary
Rearranging a formula (changing the subject) uses inverse operations to isolate a chosen variable, keeping both sides balanced. Undo operations in reverse BIDMAS order — addition/subtraction first, then multiplication/division, then powers/roots. For y = 3x + 2: subtract 2, then divide by 3, giving x = (y − 2)/3. To undo a square, take the square root; to undo a square root, square both sides (A = πr² → r = √(A/π)). For a variable in brackets, divide by the multiplier or expand first. If the new subject appears more than once, collect those terms on one side, factorise out the variable, then divide (ax = bx + c → x = c/(a − b)). The common mistakes are wrong order, not balancing both sides, incomplete rooting/squaring, failing to collect a repeated subject, and sign errors. Identify the subject, undo step by step, stay balanced, use roots and powers correctly, and factorise when needed.