GCSE Maths → Algebra → Solving Linear Equations
Solving Linear Equations GCSE Maths: Step-by-Step Guide
Solving linear equations is one of the most important GCSE algebra skills. The aim is to find the value of the unknown, usually x, that makes the equation true.
This topic connects strongly to expanding brackets, factorising, substitution, graphs, formulae, and GCSE exam problem solving.
Video explanation
A short Worthing Maths Tutor video explanation for solving linear equations GCSE maths can be embedded here later to improve student engagement and time on page.
What is a linear equation?
A linear equation is an equation where the unknown has power 1. For example:
The goal is to work out the value of x. In this example, the answer is x = 6 because 2 × 6 + 5 = 17.
Method: how to solve a linear equation
- Simplify each side if needed.
- Expand brackets if there are brackets.
- Move x terms to one side.
- Move number terms to the other side.
- Divide to find x.
- Check your answer by substituting it back in.
Example 1: Solve x + 7 = 19
We need to undo +7, so subtract 7 from both sides.
Example 2: Solve 2x + 5 = 17
First subtract 5 from both sides.
Then divide both sides by 2.
Equations with negative numbers
Negative signs often cause problems in GCSE equations. Work slowly and keep each line clear.
Example 3: Solve 5x - 6 = 14
First add 6 to both sides.
Then divide by 5.
Example 4: Solve 3x - 8 = -2
First add 8 to both sides.
Then divide by 3.
Equations with x on both sides
When x appears on both sides, collect the x terms on one side first.
Example 5: Solve 3x + 4 = x + 10
Subtract x from both sides.
Subtract 4 from both sides.
Divide by 2.
Example 6: Solve 5x - 2 = 2x + 13
Subtract 2x from both sides.
Add 2 to both sides.
Divide by 3.
Equations with brackets
If an equation contains brackets, you often need to expand the brackets first.
Example 7: Solve 3(x + 2) = 18
Expand the bracket first.
Subtract 6 from both sides.
Divide by 3.
Example 8: Solve 2(x - 3) = x + 5
Expand the bracket.
Subtract x from both sides.
Add 6 to both sides.
How to check your answer
Substitute your answer back into the original equation.
Check x = 6 in 2x + 5 = 17
The left-hand side equals the right-hand side, so x = 6 is correct.
Common mistakes in solving equations
- Doing different operations to each side.
- Changing signs incorrectly.
- Forgetting to expand brackets first.
- Losing negative signs when collecting terms.
- Dividing only one term instead of the whole side.
- Not checking the answer in the original equation.
Practice questions
Try these before checking the answers.
- Solve x + 7 = 19
- Solve 3x = 24
- Solve 2x + 9 = 21
- Solve 5x - 6 = 14
- Solve 4x + 3 = 2x + 11
- Solve 5x - 2 = 2x + 13
- Solve 3(x + 2) = 18
- Solve 2(x - 3) = x + 5
Answers
- x = 12
- x = 8
- x = 6
- x = 4
- x = 4
- x = 5
- x = 4
- x = 11
Solving linear equations FAQ
What is a linear equation?
A linear equation is an equation where the unknown has power 1, such as 2x + 5 = 17.
How do you solve a linear equation?
You solve a linear equation by using inverse operations to isolate the unknown. Whatever you do to one side, you must do to the other side.
How do you check an equation answer?
Substitute your answer back into the original equation. If both sides are equal, the answer is correct.
Why do students struggle with solving equations?
Students often struggle because of sign errors, weak negative number skills, forgetting to do the same operation to both sides, or not expanding brackets first.
Need help with GCSE algebra?
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