GCSE MATHS • HIGHER TIER
Elimination Method Made Simple
Solve simultaneous equations by eliminating one variable, solving the remaining equation and substituting back.
Clear method
Exam practice
Worked proofs
Elimination Method in Four Moves
1
Align
Write the two equations neatly underneath each other.
2
Eliminate
Add or subtract the equations to remove one variable, matching coefficients first when needed.
3
Solve
Solve the resulting equation to find the first variable.
4
Substitute
Substitute the value into an original equation to find the second variable, then check.
What You’ll Learn
Understand simultaneous equations.
Understand how elimination removes one variable.
Know when elimination is an efficient method.
Add or subtract equations correctly.
Match coefficients when necessary.
Substitute back to find the second variable.
Check solutions in the original equations.
Exact and Decimal Answers
Key Rules
Align the Equations
Write both equations directly underneath each other so the same variables line up.
Choose a Variable to Eliminate
Look for the variable that is easiest to remove.
Match the Coefficients
If necessary, multiply one or both equations so one variable has equal coefficients.
Add or Subtract
Add or subtract the equations to eliminate one variable.
Worked Proof: Elimination Method
Step-by-Step Method
- 1. Write the equations neatly underneath each other.
- 2. Identify a variable to eliminate.
- 3. Multiply equations if necessary to make coefficients equal.
- 4. Add or subtract the equations.
- 5. Solve the resulting equation.
More Worked Examples
1
Question :
2x + y = 9
3x - y = 6
Step 1: Add the equations.
(2x + y) + (3x - y) = 9 + 6
5x = 15
Step 2: Solve.
x = 3
Step 3: Substitute into the first equation.
2(3) + y = 9
6 + y = 9
y = 3
5x = 15
Step 2: Solve.
x = 3
Step 3: Substitute into the first equation.
2(3) + y = 9
6 + y = 9
y = 3
2
Question :
x + y = 10
x - y = 4
Add the equations.
2x = 14
x = 7
Substitute back.
7 + y = 10
y = 3
x = 7
Substitute back.
7 + y = 10
y = 3
3
Question :
2x + 3y = 13
2x - y = 1
Subtract the equations.
4y = 12
y = 3
Substitute back.
2x - 3 = 1
2x = 4
x = 2
y = 3
Substitute back.
2x - 3 = 1
2x = 4
x = 2
4
Question :
3x + y = 11
2x - y = 4
Add the equations.
5x = 15
x = 3
Substitute back.
y = 2
x = 3
Substitute back.
y = 2
- Common GCSE Mistakes
- Forgetting to multiply every term when matching coefficients.
- Making sign errors when adding or subtracting equations.
- Substituting incorrectly after finding the first variable.
- Failing to align equations clearly.
- Forgetting to check the final pair of values.
- Exam Tips
- Write equations directly underneath each other.
- Choose the variable that is easiest to eliminate.
- Multiply every term if you scale an equation.
- Watch signs carefully when subtracting equations.
- Substitute into an original equation after finding one variable.
- Check both values in both original equations.
Skills Used in Elimination Method
Simultaneous Equations
Elimination
Linear Equations
Substitution
Algebra
Rearranging
Checking Solutions
Algebraic Manipulation
Ready to Practise?
Printable Worksheet
Download and print practice questions with answers.
Interactive Quiz
Test your knowledge with instant feedback and hints.
Exam-Style Questions
Timed questions to build confidence for the real exam.
Watch the Elimination Method Tutorial
Step-by-step walkthroughs of Elimination Method with clear methods and exam tips.
Frequently Asked Questions
They are two equations containing the same variables, with a solution that satisfies both equations at the same time.
It adds or subtracts equations to remove one variable and create an equation with only one unknown.
It is often quickest when one pair of coefficients is already equal or can easily be made equal.
Multiply one or both equations so the coefficients of one variable become equal.
Substitute its value into one original equation to find the second variable.
Substitute both values into the original equations and confirm that both equations are satisfied.