Make x the subject: \(x + 5 = y\)
Undo +5 by subtracting 5.
Subtract 5 from both sides: \(x = y - 5\).
Rearranging Formulae
Rearranging a formula means changing the subject by applying inverse operations step by step. This skill is essential in GCSE Maths and is widely used in science-based calculations.
Overview
A formula shows a relationship between different variables.
Rearranging a formula means rewriting it so that a different letter becomes the subject.
\( V = IR \quad \Rightarrow \quad I = \frac{V}{R} \)
This topic uses the same balance idea as solving equations, but instead of finding one numerical answer, you rewrite the formula into a new form.
What you should understand after this topic
- Understand what a formula is
- Understand what the subject of a formula means
- Rearrange simple and two-step formulae
- Handle brackets and fractions correctly
- Check a rearranged formula logically
Key Definitions
Formula
An equation showing a relationship between variables.
Subject
The variable on its own on one side of the formula.
Rearrange
Rewrite the formula so a different variable becomes the subject.
Balance
Do the same operation to both sides to keep equality true.
Isolate
Get the required variable on its own.
Inverse Operation
The opposite operation, such as subtracting instead of adding.
Key Rules
Treat it like an equation
Use balance methods to isolate the required variable.
Undo operations in order
Remove additions and subtractions before multiplications and divisions when sensible.
Clear fractions carefully
Multiply both sides by the denominator if needed.
Check the final form
The required letter must be alone on one side.
How to Solve
Step 1: Understand the subject
The subject is the variable that is alone on one side of the formula.
In \(A = lw\), the subject is \(A\).
To make a new subject, you must get that variable on its own.
Exam tip: The subject should be written on the left at the end.
Step 2: Use inverse operations
Undo operations in reverse order to isolate the variable.
Addition
Undo by subtracting
Subtraction
Undo by adding
Multiplication
Undo by dividing
Division
Undo by multiplying
Step 3: Work step-by-step
Rearrange one step at a time. Keep both sides balanced.
\( y = 3x + 2 \)
Subtract 2: \( y - 2 = 3x \)
Divide by 3: \( x = \frac{y - 2}{3} \)
Exam thinking: Reverse order → subtract, then divide.
Step 4: Deal with fractions
Clear fractions first by multiplying both sides.
\( v = \frac{u + t}{m} \)
Multiply by \(m\): \( vm = u + t \)
Then rearrange normally.
Step 5: Deal with brackets
Undo multiplication outside brackets first.
\( y = a(x + b) \)
Divide by \(a\): \( \frac{y}{a} = x + b \)
Then subtract \(b\).
Step 6: Exam method summary
See solving equations for similar skills.
- Identify the subject you need.
- Undo operations in reverse order.
- Keep both sides balanced.
- Clear fractions early if needed.
- Write the final answer neatly with the subject on the left.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics, focusing on making variables the subject and applying formulas.
Edexcel
Make \( x \) the subject of \( y = x + 5 \).
Edexcel
Make \( a \) the subject of \( b = a - 7 \).
Edexcel
Make \( x \) the subject of \( y = 4x \).
Edexcel
Make \( r \) the subject of \( A = \pi r^2 \).
Edexcel
Make \( t \) the subject of \( v = u + at \).
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on algebraic manipulation and problem-solving.
AQA
Make \( x \) the subject of \( y = 3x - 4 \).
AQA
Rearrange \( y = \frac{x}{5} + 2 \) to make \( x \) the subject.
AQA
Make \( h \) the subject of \( V = \pi r^2 h \).
AQA
Rearrange \( C = 2\pi r \) to make \( r \) the subject.
AQA
A student rearranges \( y = 2x + 5 \) to \( x = 2y + 5 \).
Tick one box. Correct ☐ Incorrect ☐
Give a reason for your answer.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising reasoning, multi-step rearrangement, and fractions.
OCR
Rearrange \( 4r - p = q \) to make \( r \) the subject.
OCR
Make \( x \) the subject of \( \frac{x - 3}{5} = y \).
OCR
Make \( y \) the subject of \( ax + by = c \).
OCR
Make \( x \) the subject of \( y = \frac{3x - 2}{4} \).
OCR
Rearrange \( s = ut + \frac{1}{2}at^2 \) to make \( a \) the subject.
Exam Checklist
Step 1
Identify which variable must become the subject.
Step 2
Use balance methods, just as in solving equations.
Step 3
Undo operations in a sensible order.
Step 4
Check that the required variable is fully alone on one side.
Most common exam mistakes
Wrong subject
Rearranging correctly, but for the wrong letter.
Incomplete isolation
Leaving the required variable still multiplied or inside brackets.
Fraction errors
Forgetting brackets around the whole numerator.
One-term division mistake
Dividing only one part instead of the whole side.
Common Mistakes
These are common mistakes students make when rearranging formulae in GCSE Maths.
Not making the correct variable the subject
Incorrect
A student rearranges the formula but isolates the wrong variable.
Correct
Always check which variable the question asks you to make the subject and ensure that variable is alone on one side.
Applying different operations to each side
Incorrect
A student changes only one side of the equation.
Correct
Whatever operation you perform, you must do it to both sides of the equation to keep it balanced.
Dividing only part of an expression
Incorrect
A student divides just one term instead of the whole side.
Correct
If a bracket or expression is being divided, the entire side must be divided. Use brackets to show this clearly.
Losing brackets in fraction answers
Incorrect
A student writes a fraction without proper grouping.
Correct
Use brackets to keep the structure clear. For example, write \(\frac{a + b}{c}\), not \(a + \frac{b}{c}\).
Stopping before fully isolating the variable
Incorrect
A student leaves the variable still multiplied or inside brackets.
Correct
Continue rearranging until the required variable is completely on its own on one side of the equation.
Try It Yourself
Practise rearranging formulae to make different variables the subject.
Foundation
Foundation Practice
Rearrange simple formulae step by step.
Make x the subject: \(x - 3 = y\)
Undo -3 by adding 3.
Add 3 to both sides: \(x = y + 3\).
Make x the subject: \(2x = y\)
Undo multiplication by dividing.
Divide both sides by 2: \(x = \frac{y}{2}\).
Make x the subject: \(3x = y\)
Divide both sides by 3.
\(x = \frac{y}{3}\).
Make x the subject: \(x + 4 = 10\)
Subtract 4 from both sides.
Subtract 4: \(x = 6\).
Make x the subject: \(x/5 = y\)
Undo division by multiplying.
Multiply both sides by 5: \(x = 5y\).
Make x the subject: \(x - 7 = y\)
Undo subtraction by adding.
Add 7 to both sides: \(x = y + 7\).
Make x the subject: \(4x = 20\)
Divide both sides by 4.
\(x = 5\).
A student says from \(2x = y\), \(x = 2y\). What is wrong?
Think about undoing ×2.
To undo multiplication by 2, divide by 2: \(x = y/2\).
Make x the subject: \(x + 9 = 15\)
Subtract 9.
Subtract 9: \(x = 6\).
Higher
Higher Practice
Rearrange formulae with brackets, fractions and multiple steps.
Make x the subject: \(3x + 4 = y\)
Undo +4 first, then divide by 3.
Subtract 4: \(3x = y - 4\). Divide by 3: \(x = \frac{y - 4}{3}\).
Make x the subject: \(5x - 2 = y\)
Add 2 first, then divide.
Add 2: \(5x = y + 2\). Divide by 5: \(x = \frac{y + 2}{5}\).
Make x the subject: \(2(x + 3) = y\)
Divide by 2 first.
Divide by 2: \(x + 3 = y/2\). Subtract 3: \(x = y/2 - 3\).
Make x the subject: \(4(x - 1) = y\)
Divide first, then add 1.
Divide by 4: \(x - 1 = y/4\). Add 1: \(x = y/4 + 1\).
Make x the subject: \(\frac{x - 2}{3} = y\)
Multiply both sides by 3 first.
Multiply by 3: \(x - 2 = 3y\). Add 2: \(x = 3y + 2\).
Make x the subject: \(\frac{x + 5}{2} = y\)
Multiply by 2 first.
Multiply by 2: \(x + 5 = 2y\). Subtract 5: \(x = 2y - 5\).
Make x the subject: \(ax + b = c\)
Subtract b, then divide by a.
Subtract b: \(ax = c - b\). Divide by a: \(x = \frac{c - b}{a}\).
Make x the subject: \(px + q = r\)
Subtract q, then divide by p.
\(px = r - q\), so \(x = \frac{r - q}{p}\).
A student rearranges \(3x + 5 = y\) to \(x = y/3 + 5\). What mistake did they make?
Check the order of operations.
You must subtract 5 first, then divide. Correct: \(x = (y - 5)/3\).
Make x the subject: \(2x + 7 = 3x - 5\)
Bring x terms together first.
Subtract 2x: \(7 = x - 5\). Add 5: \(x = 12\).
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What does it mean to change the subject of a formula?
It means rearranging the formula so a different variable is on its own.
What is the main method for rearranging?
Undo operations in reverse order using inverse operations.
What common mistake should I avoid?
Forgetting to apply operations to every term in the equation.