Solving Linear Equations - AQA GCSE Mathematics Revision

Topic overview

What you need to know

Solving Linear Equations is a key part of GCSE Mathematics. Build fluency with the method, then apply it to unfamiliar and multi-step problems. An equation where the highest power of the variable is 1. It produces a straight line when graphed. E.g. 2x + 3 = 11. Undo operations in reverse order to isolate the variable. E.g. to solve 2x + 3 = 11, subtract 3 then divide by 2. Collect variable terms on one side and constants on the other. E.g. 5x − 2 = 3x + 8 → 2x = 10 → x = 5.

Exam tip

Write every stage of your method. Method marks can still be earned when the final answer is wrong.

Revision notes

Core ideas

Secure these ideas first. Say each definition in your own words, then connect it to the topic overview.

Linear Equation: An equation where the highest power of the variable is 1. It produces a straight line when graphed. E.g. 2x + 3 = 11.
Solving by Inverse Operations: Undo operations in reverse order to isolate the variable. E.g. to solve 2x + 3 = 11, subtract 3 then divide by 2.
Equations with Unknowns on Both Sides: Collect variable terms on one side and constants on the other. E.g. 5x − 2 = 3x + 8 → 2x = 10 → x = 5.
Equations with Brackets: Expand brackets first, then solve as normal. E.g. 3(x + 4) = 21 → 3x + 12 = 21 → x = 3.

Revision notes

Apply it in the exam

The exam will rarely ask for an isolated definition. Practise selecting the right idea and using it as part of a complete explanation or method.

Equations with Fractions: Multiply both sides by the denominator to eliminate fractions. E.g. x/3 = 5 → x = 15.
Forming Equations: Translate a word problem into an algebraic equation, then solve. E.g. 'Three more than twice a number is 17' → 2x + 3 = 17.
Solution / Root: The value of the variable that makes the equation true. E.g. if 2x = 10, the solution is x = 5.
Identity vs Equation: An identity (≡) is true for all values of x. An equation (=) is true only for specific values.

Core knowledge

Key facts for Solving Linear Equations

8 essentials

Linear Equation

An equation where the highest power of the variable is 1. It produces a straight line when graphed. E.g. 2x + 3 = 11.

Solving by Inverse Operations

Undo operations in reverse order to isolate the variable. E.g. to solve 2x + 3 = 11, subtract 3 then divide by 2.

Equations with Unknowns on Both Sides

Collect variable terms on one side and constants on the other. E.g. 5x − 2 = 3x + 8 → 2x = 10 → x = 5.

Equations with Brackets

Expand brackets first, then solve as normal. E.g. 3(x + 4) = 21 → 3x + 12 = 21 → x = 3.

Equations with Fractions

Multiply both sides by the denominator to eliminate fractions. E.g. x/3 = 5 → x = 15.

Forming Equations

Translate a word problem into an algebraic equation, then solve. E.g. 'Three more than twice a number is 17' → 2x + 3 = 17.

Solution / Root

The value of the variable that makes the equation true. E.g. if 2x = 10, the solution is x = 5.

Identity vs Equation

An identity (≡) is true for all values of x. An equation (=) is true only for specific values.

Active recall

Close the notes and answer these

1.Without looking, explain linear equation and give one example or consequence.
2.Without looking, explain solving by inverse operations and give one example or consequence.
3.Without looking, explain equations with unknowns on both sides and give one example or consequence.
4.Without looking, explain equations with brackets and give one example or consequence.

Content reviewed 23 July 2026 against the current linked specification.

Finished learning?

Mark the notes complete, then test what you can recall.

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