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OCR MathsAlgebra

Changing the subject of a formula

Rearrange formulae to isolate a chosen letter.

OCRGCSE MathsAlgebraFoundation and Higher
Visual model

Rearranging undoes operations in reverse order

y=3x+4y=3x+4x=y−43x=\frac{y-4}{3}subtract first, then divide
Gold-standard guide
20 mins

What you will learn

Rearrange formulae to isolate a chosen letter.
Use a clear step-by-step method for changing the subject of a formula.
Check your answer and avoid the most common exam mistake.
Useful before you start
Core number skillsEarlier algebra skillsShowing clear working
Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Method

Perform inverse operations in reverse BIDMAS order: undo addition/subtraction first, then multiplication/division, then powers/roots

Step 1

Divide both sides by pi to isolate the r^2 term

A/π\pi = r2

Step 2

Take the square root of both sides

r = A/π\sqrt{A/\pi }

Step 3

State the rearranged formula

r = A/π\sqrt{A/\pi }

Watch out

Watch out

Students multiply both sides by πi\pi instead of dividing

f
Goal

Move every term containing the subject to one side.

f
Final step

Factorise the subject, then divide by the bracket.

Worked example

Make r the subject of A = π\pi *r2.

1

Divide both sides by πt\pi to isolate the r2 term: A/π\pi = r2

2

Take the square root of both sides: r = A/π\sqrt{A/\pi }. Take the positive root since r is a length.

3

State the rearranged formula: r = A/π\sqrt{A/\pi }

Final answer

r = A/π\sqrt{A/\pi }

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Make r the subject of A = π\pi *r2.

3 marks4 minschanging-the-subject-worked
Show solution
Worked solution
  1. 1.Divide both sides by πt\pi to isolate the r2 term: A/π\pi = r2
  2. 2.Take the square root of both sides: r = A/π\sqrt{A/\pi }. Take the positive root since r is a length.
  3. 3.State the rearranged formula: r = A/π\sqrt{A/\pi }
Final answer

r = A/π\sqrt{A/\pi }

Mark points
  • M1: divide both sides by πt\pi to isolate the r2 term
  • M1: take the square root of both sides
  • M1: state the rearranged formula
  • A1: r = A/π\sqrt{A/\pi }
Watch out

Students multiply both sides by πi\pi instead of dividing.Always identify what operation the target variable is connected to and apply the inverse.Drawing an arrow diagram of operations on r can help.

Diagnosticrecall

Make v the subject of E = 12\frac{1}{2}*m*v2.

1 mark2 minschanging-the-subject-q1
Show solution
Worked solution
  1. 1.Spot the skill: Perform inverse operations in reverse BIDMAS order: undo addition/subtraction first, then multiplication/division, then powers/roots.
  2. 2.Use the divide both sides by πt\pi to isolate the r2 term stage first, then take the square root of both sides.
  3. 3.Keep the final answer visible: v = 2Em\sqrt{2\frac{E}{m}}.
Final answer

v = 2Em\sqrt{2\frac{E}{m}}

Mark points
  • M1: use the correct perform inverse operations in reverse bidmas order: undo addition/subtraction first, then multiplication/division, then powers/roots.treat the target variable like a mystery number you are unlocking step by step.
  • A1: v = 2Em\sqrt{2\frac{E}{m}}
Watch out

Students multiply both sides by πi\pi instead of dividing.Always identify what operation the target variable is connected to and apply the inverse.Drawing an arrow diagram of operations on r can help.

Easyprocedure

Make x the subject of y = 3x − 7.

2 marks3 minschanging-the-subject-q2
Show solution
Worked solution
  1. 1.Spot the skill: Perform inverse operations in reverse BIDMAS order: undo addition/subtraction first, then multiplication/division, then powers/roots.
  2. 2.Use the take the square root of both sides stage first, then state the rearranged formula.
  3. 3.Keep the final answer visible: x = (y + 7)/3.
Final answer

x = (y + 7)/3

Mark points
  • M1: use the correct perform inverse operations in reverse bidmas order: undo addition/subtraction first, then multiplication/division, then powers/roots.treat the target variable like a mystery number you are unlocking step by step.
  • A1: x = (y + 7)/3
Watch out

Students multiply both sides by πi\pi instead of dividing.Always identify what operation the target variable is connected to and apply the inverse.Drawing an arrow diagram of operations on r can help.

Mediumreasoning

Make h the subject of V = π\pi *r2*h.

3 marks4 minschanging-the-subject-q3
Show solution
Worked solution
  1. 1.Spot the skill: Perform inverse operations in reverse BIDMAS order: undo addition/subtraction first, then multiplication/division, then powers/roots.
  2. 2.Use the state the rearranged formula stage first, then divide both sides by πt\pi to isolate the r2 term.
  3. 3.Keep the final answer visible: h = V/(π\pi *r2).
Final answer

h = V/(π\pi *r2)

Mark points
  • M1: use the correct perform inverse operations in reverse bidmas order: undo addition/subtraction first, then multiplication/division, then powers/roots.treat the target variable like a mystery number you are unlocking step by step.
  • A1: h = V/(π\pi *r2)
Watch out

Students multiply both sides by πi\pi instead of dividing.Always identify what operation the target variable is connected to and apply the inverse.Drawing an arrow diagram of operations on r can help.

Hardproblem solving

Make x the subject of y = (2x + 1)/(x − 3).

3 marks5 minschanging-the-subject-q4
Show solution
Worked solution
  1. 1.Spot the skill: Perform inverse operations in reverse BIDMAS order: undo addition/subtraction first, then multiplication/division, then powers/roots.
  2. 2.Use the divide both sides by πt\pi to isolate the r2 term stage first, then take the square root of both sides.
  3. 3.Keep the final answer visible: x = (3y + 1)/(y − 2).
Final answer

x = (3y + 1)/(y − 2)

Mark points
  • M1: use the correct perform inverse operations in reverse bidmas order: undo addition/subtraction first, then multiplication/division, then powers/roots.treat the target variable like a mystery number you are unlocking step by step.
  • A1: x = (3y + 1)/(y − 2)
Watch out

Students multiply both sides by πi\pi instead of dividing.Always identify what operation the target variable is connected to and apply the inverse.Drawing an arrow diagram of operations on r can help.

Exam-stylemulti-step

Make l the subject of T = 2*π\pi *lg\sqrt{\frac{l}{g}}.

4 marks6 minschanging-the-subject-q5
Show solution
Worked solution
  1. 1.Spot the skill: Perform inverse operations in reverse BIDMAS order: undo addition/subtraction first, then multiplication/division, then powers/roots.
  2. 2.Use the take the square root of both sides stage first, then state the rearranged formula.
  3. 3.Keep the final answer visible: l = g*T2/(4*π\pi 2).
Final answer

l = g*T2/(4*π\pi 2)

Mark points
  • M1: use the correct perform inverse operations in reverse bidmas order: undo addition/subtraction first, then multiplication/division, then powers/roots.treat the target variable like a mystery number you are unlocking step by step.
  • A1: l = g*T2/(4*π\pi 2)
Watch out

Students multiply both sides by πi\pi instead of dividing.Always identify what operation the target variable is connected to and apply the inverse.Drawing an arrow diagram of operations on r can help.

Grade 9 stretchproblem solving

Make x the subject of y = (3x - 2) / (x + 4).

4 marks7 minssubject-g9
Show solution
Worked solution
  1. 1.Multiply both sides by x + 4.
  2. 2.Collect the x terms on one side.
  3. 3.Factorise and divide.
Final answer

x = (4y + 2) / (3 - y)

Mark points
  • M1: yx + 4y = 3x - 2
  • M1: x(y - 3) = -2 - 4y
  • A1: x = (4y + 2)/(3 - y)
Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint
12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Changing the subject of a formula - 2 marksMake v the subject of E = 12\frac{1}{2}*m*v2.Mark answer
Answer

v = 2Em\sqrt{2\frac{E}{m}}

2Algebraic notation and substitution - 2 marksFind t² + 4t when t = −2Mark answer
Answer

−4

3Simplifying expressions - 2 marksSimplify 2xy + 3x²y − xy + x²yMark answer
Answer

3x²y + xy

4Expanding and factorising - 3 marksFactorise fully 6x² + 9xMark answer
Answer

3x(2x + 3)

Mastery check
  • I can explain the method for changing the subject of a formula.
  • I can show clear working without skipping key steps.
  • I can avoid this mistake: Students multiply both sides by πi\pi instead of dividing.Always identify what operation the target variable is connected to and apply the inverse.Drawing an arrow diagram of operations on r can help.
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Explain changing the subject of a formula from the beginning.Give me a hint for a changing the subject of a formula question.Quiz me on changing the subject of a formula.
Related topics
Algebraic notation and substitutionSimplifying expressionsExpanding and factorising
Official exam-board sources

This guide follows the OCR GCSE Mathematics J560 specification. Practice questions are original Learnova questions shaped around official content and exam skills.

Official specification Official past papers
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