Key ideas
- The subject of a formula is the letter on its own on one side. In A = lb, the subject is A.
- Changing the subject means rearranging the formula so a different letter is on its own. It works just like solving an equation: do the same to both sides.
- Undo the operations in reverse order, one step at a time. Deal with anything added or taken away first, then multiplying and dividing, then squares and roots.
- Undo + with −, undo × with dividing, undo dividing with ×, undo a square with √, and undo √ by squaring.
- When you divide a whole side by something, the whole side goes on top of the fraction: y − c = mx gives x = y − cm.
- When you square root, the root goes over the whole side: r² = Aπ gives r = Aπ.
- Write each step on a new line. Exam questions are usually 2 or 3 marks, with one mark for each correct step.
Typing answers: type a fraction with a slash and brackets, like x = (y - c)/m. Use the √ and π buttons, or type sqrt and pi. Check the preview under the box: it shows how your answer is being read.
One and two steps
Change the subject of the formula y = mx + c to x.(2 marks)
Undo the + c.y − c = mx take c from both sides✓1
In plain EnglishThe x is multiplied by m and then c is added. Undo in reverse order: take c away from both sides first.
Undo the × m.y − cm = x divide both sides by mx = y − cm✓2
In plain Englishx is multiplied by m, so divide both sides by m. The whole of y − c goes on top of the fraction. Then write it with x first.
⚠ A common mistake: writing x = y − c/m. That only divides the c by m. The whole of y − c must be divided, so it all goes on top.
Fractions
Change the subject of the formula M = a + b2 to a.(2 marks)
Clear the fraction.2M = a + b multiply both sides by 2✓1
In plain EnglishThe right-hand side is divided by 2. Undo it by multiplying both sides by 2.
Undo the + b.a = 2M − b✓2
In plain EnglishTake b away from both sides, then write it with a first.
Squares and square roots
Change the subject of the formula A = πr² to r.(2 marks)
Undo the × π.Aπ = r² divide both sides by π✓1
In plain Englishr is squared and then multiplied by π. Undo the multiplying first: divide both sides by π.
Undo the square.r = Aπ✓2
In plain EnglishSquare root both sides. The root goes over the whole fraction.
Change the subject of the formula v = 2gh to h.(2 marks)
Undo the square root.v² = 2gh square both sides✓1
In plain Englishh is inside a square root. Undo it by squaring both sides.
Undo the × 2g.h = v²2g✓2
In plain Englishh is multiplied by 2g, so divide both sides by 2g.
Brackets and negatives
Change the subject of the formula P = 2(l + b) to b.(2 marks)
Undo the × 2.P2 = l + b divide both sides by 2✓1
In plain EnglishThe whole bracket is multiplied by 2, so divide both sides by 2. (You could also multiply out the bracket first: P = 2l + 2b.)
Undo the + l.b = P2 − l✓2
In plain EnglishTake l away from both sides.
Also right: b = P − 2l2. Different-looking answers can be equal. The checker on this page accepts any correct form.
Change the subject of the formula y = 5 − 3x to x.(2 marks)
Move the x term to make it positive.3x + y = 5 add 3x to both sides3x = 5 − y take y from both sides✓1
In plain EnglishAdd 3x to both sides so the x term is positive. Then take y from both sides.
Undo the × 3.x = 5 − y3✓2
In plain EnglishDivide both sides by 3.
⚠ Watch the signs: x = y − 53 is wrong. If you divide by −3 instead, you get x = y − 5−3, which is the same as 5 − y3.
Subject on the bottom
Change the subject of the formula T = km to m.(2 marks)
Get m off the bottom.Tm = k multiply both sides by m✓1
In plain Englishm is on the bottom of the fraction. Multiply both sides by m to get it off the bottom.
Undo the × T.m = kT✓2
In plain EnglishDivide both sides by T.
Exam-style questions
Change the subject of the formula s = ut + 12at² to a.(3 marks)
Undo the + ut.s − ut = 12at² take ut from both sides✓1
In plain EnglishThe a is in the last term. Take ut away from both sides to leave that term on its own.
Undo the half.2(s − ut) = at² multiply both sides by 2✓2
In plain EnglishMultiply both sides by 2. Put s − ut in a bracket: all of it is multiplied by 2.
Undo the × t².a = 2(s − ut)t²✓3
In plain EnglishDivide both sides by t².
Also right: a = 2s − 2utt².
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Your turn
Type your answer with the new subject first, like x = (y - c)/m. Use brackets round a whole top or bottom, and the √, π and ² buttons. The preview shows how your answer is read. Any correct form is accepted. Stuck? Tap Need a hint?
Past paper questions
Ready for exam-style questions? Changing the subject comes up in National 5 almost every year, usually for 2 or 3 marks.
National 5 changing the subjectEvery National 5 changing-the-subject question in one PDF (2014 to 2025), with answers.maths.scot · PDF Open ↗ (opens in a new tab) Changing the subject: topic pageEach question separately, with worked videos.maths.scot Open ↗ (opens in a new tab) Changing the subject: past paper questionsPast paper questions on changing the subject of a formula.Dynamic Maths Open ↗ (opens in a new tab)