Algebra · GCSE Maths
Changing the subject of a formula
GCSE Maths changing the subject: use inverse operations in reverse order, including squares, fractions and brackets, and factorise when the new subject appears twice.
A formula is an equation. To make a letter the subject, undo the operations in reverse order, doing the same to both sides, until that letter is alone on the left.
The important bits
What you need to know
- 1
The subject is the letter sitting alone on one side, usually the left: C = 2πr has C as subject. Making r the subject means r = C / (2π).
- 2
Use inverse operations in reverse order, as when solving for x. If v was multiplied by t and then u was added, subtract u first, then divide by t: s = ut + ½at² is heavier, but the same idea.
- 3
Whatever you do to one side, do to the other. Write a new formula after every step so the examiner can follow the balance.
- 4
Roots and squares: if A = πr², divide by π, then square-root: r = √(A/π). The GCSE principal square root is non-negative for a length.
- 5
Fractions: if 1/f = 1/u + 1/v, a common first move is to combine the right, or to take reciprocals with care. Clearing denominators by multiplying through is often safer.
- 6
If the new subject appears in two terms, collect those terms and factorise. From ax + b = cx + d, ax − cx = d − b, x(a − c) = d − b, x = (d − b)/(a − c).
- 7
Brackets: expand or divide to remove them. 2(c + 5) = d, divide by 2, c + 5 = d/2, c = d/2 − 5. Do not divide only the c by 2.
- 8
Check by substituting a simple number. If C = 10π and you claim r = C / (2π), then r = 5, and 2π × 5 = 10π recovers C.
Quotations worth analysing
Short evidence. Real method.
“Make r the subject of the formula”
The answer must start r = … . Leaving r on both sides, or leaving a ± that the context forbids for a length, loses the final mark.
“Factorise so that x appears only once”
You cannot divide by x if x is still sitting in another term. Collect x terms, factor, then divide. That three-line pattern is the method.
“Square root both sides”
If r² = A/π, then r = √(A/π) for a length. Writing r = A/π without the root, or rooting only A, are the two common incomplete inverses.
Go deeper
Reverse order, one operation per line
v = u + at, make a the subject. a is multiplied by t, then u is added. Undo addition first: v − u = at. Then divide by t: a = (v − u)/t. Students who divide v by t first have broken the formula. The same reverse-order idea runs A = πr²: divide by π, A/π = r², then square-root. It runs V = 4/3 πr³: multiply both sides by 3/4, divide by π, then cube-root. Write the inverse in words in revision (“subtract u, divide by t”) until the order is automatic. In the exam, the words can disappear but the lines must remain: each line a true equation with a single change. That is how a sign error still collects method marks.
Go deeper
The subject in two places needs factorising
Make x the subject of y = (x + 3)/(x − 1). Multiply both sides by (x − 1): y(x − 1) = x + 3. Expand: yx − y = x + 3. Collect x terms: yx − x = 3 + y. Factor: x(y − 1) = 3 + y. Divide: x = (3 + y)/(y − 1), for y ≠ 1. The factorising line is the one students skip, then they try to “cancel x” while x is still added on the other side. Another classic: make t the subject of v = t/(1 + t). Then v + vt = t, vt − t = −v, t(v − 1) = −v, t = v / (1 − v). Check with a number: if t = 1, v = 1/2, and (1/2) / (1 − 1/2) = 1, which recovers t.
Go deeper
Fractions, squares and the answer line
1/R = 1/P + 1/Q, make P the subject. Subtract 1/Q: 1/R − 1/Q = 1/P. Combine the left: (Q − R)/(RQ) = 1/P. Take reciprocals: P = RQ / (Q − R). Alternatively multiply through by PQR at the start to clear denominators, then collect P. Both routes are fine; mixing them halfway is not. For a square, decide whether the context is a length (take the positive root) or a full solve (include ± if the paper is doing equations rather than formulae for sides). The answer line wants P = … with P appearing once. If P is still in a denominator on the right, you have not finished. Expand and factor until the letter sits by itself with a coefficient of 1.
See the idea in action
Make r the subject of A = 4πr², then make x the subject of y = 2x + 3x. First formula: Step 1: Divide both sides by 4π: A / (4π) = r². Step 2: Square-root both sides: r = √(A / (4π)). (Length, so the positive root.) Second formula: Step 3: y = 2x + 3x is y = 5x after collecting like terms. Step 4: Divide both sides by 5: x = y/5. A harder check of the two-term idea: make x the subject of ax + b = cx. Step 5: ax − cx = −b, then x(a − c) = −b, so x = −b / (a − c) = b / (c − a).
Exam technique
Turn knowledge into marks
If the new subject appears twice, stop dividing. Collect those terms, factor the letter out, then divide. The factorising line is what the mark scheme is looking for.
Common mistakes
Do not give these marks away
- 01
Dividing only one term by a number, such as turning 2(c + 5) = d into c + 5 = 2d or into 2c + 5 = d.
- 02
Leaving the new subject on both sides because it appeared twice, without collecting and factorising.
- 03
Square-rooting only the top of a fraction, writing r = √A / π instead of r = √(A/π) from A = πr².
C = 2πr rearranged to make r the subject is
Ar = C / (2π)
Br = 2πC
Cr = C / 2 + π
Dr = 2C / π
Show the answer
r = C / (2π). Divide both sides by 2π. r = 2πC multiplies instead of dividing. r = C / 2 + π only divided by 2 and then added π. r = 2C / π inverted the 2 and the π inconsistently.
Quick questions
If this is the bit you searched
How do you change the subject of a formula GCSE?
Use inverse operations in reverse order, doing the same to both sides, until the required letter is alone. Write a new formula after each step.
What if the letter appears twice?
Collect the terms that contain it, factorise that letter out, then divide both sides by the remaining bracket. You cannot cancel it while it still sits in two places.
How do I deal with a square in the formula?
Isolate the square first, then square-root both sides. For a length, take the positive root. For A = πr², r = √(A/π).
Is changing the subject the same as solving an equation?
Yes: a formula is an equation with more than one letter. The balance method is identical. The answer is another formula, not a number, unless values are given.