GCSE Maths · Algebra

Linear equations GCSE Questions and Worked Answers

To solve a linear equation, find the value that makes both sides equal. Undo operations on both sides, simplify carefully, and check the value in the original equation.

Foundation & Higher6 worked examples10 original questions

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Start with the meaning

What you need to know about linear equations

Three identical tickets and a £2 booking fee cost £17. Call the unknown ticket price x. The statement 3x + 2 = 17 says that these two ways of describing the cost are equal; 3x means three times x. Solving the equation means finding the ticket price that makes that statement true. Removing the same £2 from both descriptions leaves 3x = 15, so one ticket costs £5.

See the idea first

Keep the two totals equal

An equation has an equals sign. Each side is a description of the same amount. Doing the same operation to both sides preserves that equality.

Whole purchase3x + 2 = 17three tickets plus one fee

Subtract 2 from both sides3x = 15only the tickets remain

Divide both sides by 3x = 5one ticket is one third of £15

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Undo two operations

What the problem asks: Solve 4x − 3 = 21.

How to solve it: Add 3 to both sides to get 4x = 24, then divide both sides by 4: x = 6.

Solve with x on both sides

What the problem asks: Solve 5x + 2 = 2x + 14.

How to solve it: Subtract 2x from both sides, then subtract 2. This leaves 3x = 12, so x = 4.

Solve a bracketed equation

What the problem asks: Solve 3(x + 2) = 24.

How to solve it: Here the whole left side is a multiple of 3, so divide both sides by 3 first: x + 2 = 8. Subtract 2 to get x = 6. Expanding first also works.

Solve with a fraction

What the problem asks: Solve x/4 + 2 = 7.

How to solve it: Subtract 2 to get x/4 = 5, then multiply both sides by 4: x = 20.

A reliable routine

For equations that simplify to ax + b = cx + d

These are linear equations in x: after simplifying, x appears only to the first power. The balancing method works because adding or subtracting the same amount, or multiplying or dividing by the same non-zero number, preserves equality.

Simplify each side if useful: expand brackets and collect like terms.
Add or subtract the same x term on both sides to gather the unknown on one side.
Undo the remaining addition or subtraction on both sides.
Divide by the non-zero coefficient of x, then substitute the answer into the original equation.

Check: Do not describe a term as jumping across the equals sign. Show the operation on both sides. If all x terms cancel, check whether the remaining statement is always true or impossible.

Fully worked

Linear equations GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Two operations

2 marks

Question

Solve 6x + 5 = 29.

Subtract 5 from both sides.

6x=24

Divide both sides by 6.

x=4

Check: 6 × 4 + 5 = 29.

Example 2

A negative coefficient

2 marks

Question

Solve 7 − 3x = 22.

Subtract 7 from both sides.

-3x=15

Divide by −3, not 3.

x=-5

Check: 7 − 3(−5) = 22.

Example 3

Unknown on both sides

3 marks

Question

Solve 7x − 4 = 3x + 20.

Subtract 3x from both sides.

4x-4=20

Add 4 to both sides.

4x=24

x=6

Both original sides now equal 38.

Example 4

Expand before collecting

3 marks

Question

Solve 2(3x − 4) = x + 17.

Multiply both terms in the bracket by 2.

6x-8=x+17

Subtract x, then add 8, on both sides.

5x-8=17

5x=25

x=5

The original sides both equal 22.

Example 5

A whole numerator

3 marks

Question

Solve $\dfrac{x+3}{4}=5$ .

The whole x + 3 is divided by 4. Multiply both sides by 4.

x+3=20

Subtract 3.

x=17

Check: (17 + 3)/4 = 5.

Example 6

Build the equation

4 marks

Question

A rectangle is x cm wide and (x + 5) cm long. Its perimeter is 34 cm. Find its dimensions.

Perimeter counts two widths and two lengths.

2x+2(x+5)=34

4x+10=34

4x=24

x=6

The width is 6 cm and length is 11 cm. Check: 6 + 11 + 6 + 11 = 34.

10 original questions · total 23 marks

Linear equations GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 28 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed

One step

1 mark

Solve x + 9 = 16.

Show worked answer

Subtract 9 from both sides.

x=7

Check: 7 + 9 = 16.

Two steps

2 marks

Solve 5x − 6 = 24.

Show worked answer

Add 6 to both sides. $5x=30$ Divide both sides by 5. $x=6$

Negative answer

2 marks

Solve 4x + 7 = −9.

Show worked answer

Subtract 7.

4x=-16

Divide by 4.

x=-4

Negative multiplier

2 marks

Solve 10 − 2x = 18.

Show worked answer

Subtract 10.

-2x=8

Divide by −2.

x=-4

Check: 10 − 2(−4) = 18.

Both sides

3 marks

Solve 8x + 1 = 3x + 26.

Show worked answer

Subtract 3x.

5x+1=26

Subtract 1, then divide by 5.

5x=25

x=5

Brackets

3 marks

Solve 4(x − 2) = 2x + 10.

Show worked answer

4x-8=2x+10

Subtract 2x, then add 8.

2x=18

x=9

Both original sides equal 28.

Fraction

2 marks

Solve $x/5-3=4$ .

Show worked answer

Add 3, then multiply by 5.

x/5=7

x=35

Two denominators

Harder3 marks

Solve $\dfrac{x}{3}+\dfrac{x}{2}=10$ .

Show worked answer

Multiply every term on both sides by 6 to remove both denominators.

2x+3x=60

5x=60

x=12

Check: 4 + 6 = 10.

No solution

Harder2 marks

Solve 3(x + 2) = 3x + 5, or explain why there is no solution.

Show worked answer

3x+6=3x+5

Subtract 3x from both sides.

6=5

This is impossible for every x, so there is no solution. It does not mean x = 0.

Ticket context

3 marks

Four identical tickets and a £3 fee cost £31. Find the price of one ticket.

Show worked answer

Let x be the ticket price in pounds.

4x+3=31

4x=28

x=7

Each ticket costs £7; 4 × £7 + £3 = £31.

Examiner-style feedback

Common linear equations mistakes

Changing only one side

An equation stays balanced only when you apply the same operation to both sides.

Losing a minus sign

In 7 − 3x, the coefficient of x is −3. Keep the sign when dividing.

Multiplying just one fraction

To clear denominators, multiply every term on both sides by the common multiple.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

A solution makes the original equality true.
Balance both sides at every step.
Expand and collect only where useful.
Check by substitution.

Quick answers

Linear equations FAQ

Must I always expand brackets first?

No. If the whole side is multiplied by one number, dividing first can be shorter. Both routes are valid if equality is preserved.

Can a linear equation have infinitely many solutions?

Yes. For example, 2(x + 1) = 2x + 2 simplifies to 2 = 2 after subtracting 2x, so every real value of x works.

Build connected skills

What to revise next

Prepare the expressions

Expanding brackets

Understand distribution before solving.

Revise expanding brackets

Keep the letters

Rearranging formulae

Find a rule for a different unknown.

Revise rearranging formulae

Content standards

Curriculum and rights review

A17: linear equations in one unknown, including unknowns on both sides, brackets and numerical denominators, across tiers. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references

AQA GCSE Maths subject content