Solve two-step equations, equations with brackets and equations with the unknown on both sides using the balance method. It is on Foundation and Higher, aimed at grade 3. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Two-step and brackets equations still use the balance idea, with more than one inverse. Undo addition and subtraction first, then multiplication and division — the reverse of BIDMAS. 2x+7=19 becomes 2x=12 then x=6. Expand brackets before collecting if both sides have them.
Unknowns on both sides: collect the x terms on one side and the numbers on the other. 5x−3=2x+9 becomes 3x=12. If the x coefficient would be negative, you can add extra x to both sides to keep it positive. Always substitute back.
Undo one operation per line, both sides. Expand brackets first (or divide by the number outside if it goes into both sides). Letters on both sides: collect the x terms on the side that stays positive if you can.
Objectives
In 2x + 3 = 11, subtract 3, then divide by 2.
Whatever you do to one side, do to the other.
In 3x + 2 = x + 10, subtract x first so the letter is on one side.
Key points
Undo operations in reverse order, doing the same to both sides.
Expand brackets before solving.
Collect the unknowns on one side when they appear on both.
Handle fractions by multiplying both sides by the denominator.
Worked examples
Example 1
Solve 2x + 3 = 7.
2 marks
Start from 2x + 3 = 7.
Undo the add or subtract, then divide by the number in front of x.
The result is 2.
Answer: 2
Common slips
Slip: Start from 4x + 3 = 7. That changes a number from the question, so it answers a different question.
Slip: Skip this step: Undo the add or subtract, then divide by the number in front of x. Every step in the working is needed. Skipping one loses the method.
Slip: The result is 6. That does not match the working above it.
Example 2
Solve 6x + 2 = 2x + 6.
2 marks
Collect the x terms from 6x + 2 = 2x + 6.
Divide by the coefficient of x.
The result is 1.
Answer: 1
Common slips
Slip: Collect the x terms from 8x + 2 = 2x + 6. That changes a number from the question, so it answers a different question.
Slip: Skip this step: Divide by the coefficient of x. Every step in the working is needed. Skipping one loses the method.
Slip: The result is 5. That does not match the working above it.
Example 3
Solve 2(x + 3) = 8.
3 marks
Expand the bracket: 2(x + 3) = 2x + 6.
Take 6 from both sides: 2x = 2.
Divide both sides by 2: x = 1.
Answer: 1
Common slips
Slip: Expand the bracket: 4(x + 3) = 2x + 6. That changes a number from the question, so it answers a different question.
Slip: Take 9 from both sides: 2x = 2. That changes a number in the working, so the step no longer follows from the one before.
Slip: Divide both sides by 6: x = 1. That does not match the working above it.
Questions and answers
GCSE Maths solving linear equations questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Solve 2x − 1 = 9.
Show answer
Answer: 5
Start from 2x − 1 = 9.
Undo the add or subtract, then divide by the number in front of x.
M1 — Start from 2x − 1 = 9.
A1 — Answer 5
Question 2 Easy2 marks
A number is multiplied by 3, then 4 is added. The result is 13. What is the number?
Show answer
Answer: 3
Start from 3x + 4 = 13.
Undo the add or subtract, then divide by the number in front of x.
M1 — Start from 3x + 4 = 13.
A1 — Answer 3
Question 3 Medium2 marks
Solve 3(x + 4) = 18.
Show answer
Answer: 2
Expand the bracket: 3(x + 4) = 3x + 12.
Take 12 from both sides: 3x = 6.
M1 — Expand the bracket: 3(x + 4) = 3x + 12.
A1 — Answer 2
Question 4 Medium2 marks
Solve 3x − 2 = 16.
Show answer
Answer: 6
Start from 3x − 2 = 16.
Undo the add or subtract, then divide by the number in front of x.
M1 — Start from 3x − 2 = 16.
A1 — Answer 6
Question 5 Hard2 marks
Solve 8x + 4 = 4x + 24.
Show answer
Answer: 5
Collect the x terms from 8x + 4 = 4x + 24.
Divide by the coefficient of x.
M1 — Collect the x terms from 8x + 4 = 4x + 24.
A1 — Answer 5
Question 6 Hard2 marks
Solve 4(x + 5) = 32.
Show answer
Answer: 3
Expand the bracket: 4(x + 5) = 4x + 20.
Take 20 from both sides: 4x = 12.
M1 — Expand the bracket: 4(x + 5) = 4x + 20.
A1 — Answer 3
Question 7 Exam-style3 marks
Solve 5x + 9 = 34.
Show answer
Answer: 5
Start from 5x + 9 = 34.
Undo the add or subtract, then divide by the number in front of x.
The result is 5.
M1 — Start from 5x + 9 = 34.
M1 — Undo the add or subtract, then divide by the number in front of x.
A1 — Answer 5
Common mistakes
Most lost marks on this topic are one slip.
Dividing by 2 before subtracting 3 in 2x + 3 = 11.
Undo the add first, because it was done last.
Subtracting x from one side only.
Take x from both sides.
Checking with a different equation from the one you solved.
Use the original equation.
Spot the slip
The inverse of × 2 then + 3 starts with
− 3
÷ 2
+ 3
Show the slip
− 3 Undo the last operation first.
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, with the equals signs lined up — clear working earns method marks even if you slip.
Always write the final line as x= … .
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.