What is changing the subject?
Changing the subject of a formula means rearranging it so that a different variable is isolated on one side. For example, turning v = u + at into a = (v − u)/t makes a the subject.
Use inverse operations — exactly as when solving an equation — and work in this order:
- Remove additions and subtractions first
- Remove multiplications and divisions
- Remove powers and roots last
- If the required subject appears in two terms, collect those terms together and factorise
Linear rearrangement
Worked Example 1
Make a the subject of: v = u + at
1
Subtract u from both sides: v − u = at
2
Divide both sides by t: a = (v − u) / t
Answera = (v − u) / t
Powers and roots
Worked Example 2
Make r the subject of: A = πr²
1
Divide both sides by π: A/π = r²
2
Square root both sides: r = √(A/π)
Answerr = √(A / π)
Worked Example 3 — Square and root together
Make l the subject of: T = 2π√(l/g)
1
Divide both sides by 2π: T / 2π = √(l/g)
2
Square both sides: (T / 2π)² = l/g
3
Multiply by g: l = gT² / 4π²
Answerl = gT² / 4π²
Subject appearing twice
When the required subject appears in two different terms, collect those terms on one side, then factorise out the subject.
Worked Example 4
Make x the subject of: y = (x + 3) / (x − 1)
1
Multiply both sides by (x − 1): y(x − 1) = x + 3
2
Expand: yx − y = x + 3
3
Collect x terms on one side: yx − x = y + 3
4
Factorise: x(y − 1) = y + 3
5
Divide: x = (y + 3) / (y − 1)
Answerx = (y + 3) / (y − 1)
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1Make x the subject of: y = 3x + 5Foundation
Q2Make a the subject of: v = u + atFoundation
Q3Make r the subject of: A = πr²Foundation
Q4Make l the subject of: P = 2(l + w)Foundation
Q5Make h the subject of: A = ½bhFoundation
Higher (Grade 5–7)
Q6Make C the subject of: F = 9C/5 + 32Higher
Q7Make s the subject of: v² = u² + 2asHigher
Q8Make m the subject of: y = mx + cHigher
Q9Make a the subject of: s = ½n(a + l)Higher
Q10Make l the subject of: T = 2π√(l/g)Higher
Higher — Hard (Grade 8–9)
Q11Make x the subject of: y = (x + 3) / (x − 1)Grade 8–9
Q12Make u the subject of: v = √(u² + 2as)Grade 8–9
Q13Make u the subject of: 1/f = 1/u + 1/vGrade 8–9
Q14Make x the subject of: y = ax² − bGrade 8–9
Q15Make x the subject of: ax + b = cx + dGrade 8–9
Answers
Foundation (Q1–Q5)
Q1x = (y − 5) / 3(subtract 5, divide by 3)
Q2a = (v − u) / t(subtract u, divide by t)
Q3r = √(A / π)(divide by π, square root)
Q4l = P/2 − w(divide by 2, subtract w)
Q5h = 2A / b(multiply by 2, divide by b)
Higher (Q6–Q10)
Q6C = 5(F − 32) / 9(subtract 32, multiply by 5, divide by 9)
Q7s = (v² − u²) / 2a(subtract u², divide by 2a)
Q8m = (y − c) / x(subtract c, divide by x)
Q9a = 2s/n − l(multiply by 2, divide by n, subtract l)
Q10l = gT² / 4π²(divide by 2π, square both sides, multiply by g)
Higher — Hard (Q11–Q15)
Q11x = (y + 3) / (y − 1)(multiply out, collect x terms, factorise: x(y−1) = y+3)
Q12u = √(v² − 2as)(square both sides: v²=u²+2as; u²=v²−2as; square root)
Q13u = fv / (v − f)(1/u = 1/f − 1/v = (v−f)/fv; invert: u = fv/(v−f))
Q14x = √((y + b) / a)(add b, divide by a, square root)
Q15x = (d − b) / (a − c)(ax − cx = d − b; x(a−c) = d−b; divide)
Common mistakes
Common Mistake 1
Applying operations to only one side
Every operation must be applied to both sides of the equation. A common error is adding or subtracting a term from one side only. Write out each step clearly and check both sides balance.
Common Mistake 2
Wrong order — dividing before removing addition
For y = 3x + 5, students sometimes divide by 3 first to get y/3 = x + 5/3. Whilst technically valid, it leads to messy fractions. Work in the standard order: subtract 5 first, then divide by 3 to get x = (y − 5)/3.
Common Mistake 3
Square-rooting individual terms instead of the whole expression
From r² = A/π, some students write r = √A/π instead of r = √(A/π). The square root applies to the entire right-hand side. Use brackets: r = √(A/π).
Common Mistake 4
Forgetting to factorise when the subject appears twice
In y(x − 1) = x + 3, after expanding to yx − y = x + 3, students forget to collect x terms. The key step is yx − x = y + 3, then x(y − 1) = y + 3, then divide. Missing the factorise step makes it impossible to isolate x.
Common Mistake 5
Sign errors when multiplying out a bracket
When multiplying y(x − 1), students sometimes write yx − 1 instead of yx − y. The y multiplies every term inside the bracket, including the −1.
Want to improve your grade faster?
If changing the subject is still causing problems, Alamin's diagnostic approach identifies exactly which skills are missing and builds a targeted plan to address them — with AI-powered practice between sessions.
Book an Assessment Session (£60)No upfront payment required — payment is taken after confirmation.