Make a letter the subject when it appears twice, sits inside a fraction, or is under a square root or squared. It is Higher only, aimed at grade 7. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Harder rearrangement puts the new subject in two places, under a root, or in a denominator. Factorise the subject out: x(a+b)=…. Multiply through by a denominator to clear a fraction. Square both sides to undo a root (and note if extra solutions might appear — at GCSE the formula is usually one-way). The subject should appear once, on its own, on the left.
Do the inverse operations in a sensible order: fractions first (multiply up), then add/subtract, then factorise, then divide. Check with numbers: pick values for the other letters, compute both versions, they must agree.
When the subject is on both sides, collect those terms. When it is inside a fraction, multiply through. Square to undo a square root (and consider the sign). Factor the subject out if it sits in two terms.
Objectives
Collect x when it is on both sides.
Undo a square root or a square.
Make x the subject of a bracket such as y(x-2) = 3.
Key points
Clear fractions by multiplying both sides by the denominator.
Expand brackets and gather every term containing the new subject on one side.
Factorise out the subject when it appears more than once.
Undo a square root by squaring both sides, and a square by square-rooting (with ± where appropriate).
Worked examples
Example 1
Make x the subject of y=2x+6.
2 marks
Subtract 6 from both sides: y - 6 = 2x.
Divide both sides by 2: x = (y - 6)/2.
State rearranged subject: x = (y-6)/2.
Answer: (y-6)/2
Common slips
Slip: Divide by 2 first: y/2 = x + 6. Undo addition before division.
Slip: Multiply both sides by 2 prematurely. Division is the inverse of multiplication.
Slip: Write y/2 - 6. The entire numerator (y - 6) is divided by 2.
Example 2
Make x the subject of y=√(x+1).
2 marks
Square both sides to eliminate the square root: y² = x + 1.
Subtract 1 from both sides: x = y² - 1.
State the formula: x = y²-1.
Answer: y²-1
Common slips
Slip: Subtract 1 first under the root. You must undo the outer square root first.
Slip: Add 1 to both sides. Subtracting 1 isolates x.
Slip: Write x = y+1. Squaring y gives y².
Example 3
Make x the subject of 5x+2=yx+8.
3 marks
Rearrange all x terms to one side: 5x - yx = 8 - 2.
Factor out x: x(5 - y) = 6.
Divide by (5 - y): x = 6/(5-y).
Answer: 6/(5-y)
Common slips
Slip: Divide by y immediately. Gather terms with the target subject x first.
Slip: Combine to 5yx = 6. Factorisation isolates a single instance of x.
Slip: Multiply by (5 - y). Dividing by the bracket leaves x as the subject.
Questions and answers
GCSE Maths harder rearranging formulae questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Make x the subject of y=3x+7.
Show answer
Answer: (y-7)/3
Subtract 7: y - 7 = 3x.
Divide by 3: x = (y-7)/3.
M1 — Subtract 7: y - 7 = 3x.
A1 — Divide by 3: x = (y-7)/3.
Question 2 Easy2 marks
Make x the subject of y=4x+1.
Show answer
Answer: (y-1)/4
Subtract 1: y - 1 = 4x.
Divide through by 4: x = (y-1)/4.
M1 — Subtract 1: y - 1 = 4x.
A1 — Divide through by 4: x = (y-1)/4.
Question 3 Medium2 marks
Make x the subject of y=√(x+4).
Show answer
Answer: y²-4
Square: y² = x + 4.
Subtract 4: x = y² - 4.
M1 — Square: y² = x + 4.
A1 — Subtract 4: x = y² - 4.
Question 4 Medium2 marks
Make x the subject of y=√(x+9).
Show answer
Answer: y²-9
Square: y² = x + 9.
Subtract 9: x = y² - 9.
M1 — Square: y² = x + 9.
A1 — Subtract 9: x = y² - 9.
Question 5 Hard2 marks
Make x the subject of 4x+1=yx+9.
Show answer
Answer: 8/(4-y)
4x - yx = 8.
x(4 - y) = 8, so x = 8/(4-y).
M1 — 4x - yx = 8.
A1 — x(4 - y) = 8, so x = 8/(4-y).
Question 6 Hard2 marks
Make x the subject of 3x+2=yx+8.
Show answer
Answer: 6/(3-y)
3x - yx = 6.
x(3 - y) = 6, so x = 6/(3-y).
M1 — 3x - yx = 6.
A1 — x(3 - y) = 6, so x = 6/(3-y).
Question 7 Exam-style4 marks
Make x the subject of 6x+3=yx+11.
Show answer
Answer: 8/(6-y)
Rearrange terms with x to one side: 6x - yx = 11 - 3.
Simplify constants: 6x - yx = 8.
Factorise x: x(6 - y) = 8.
Divide by (6 - y): x = 8/(6 - y).
State subject formula: 8/(6-y).
M1 — Collect x terms on one side and constants on the other: 6x - yx = 11 - 3
M1 — Factorise x: x(6 - y) = 8
M1 — Divide by (6 - y)
A1 — 8/(6-y)
Common mistakes
Most lost marks on this topic are one slip.
Dividing by y when x is also in yx
Collect every x term first
Undoing a square root by dividing
Square both sides
Subtracting 2 instead of adding it back from y(x-2)
After dividing, add the 2
Spot the slip
y = √(x+1). Pat writes x = √y - 1. What is the correct x?
y²-1
y²+1
√y-1
Show the slip
y²-1 Square both sides, then subtract 1.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Write one operation per line and say what you did ('× both sides by (x−3)') — markers follow it easily and award method marks.
Underline or circle the subject in every line so you can see where it is.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.