GCSE Maths • Foundation & Higher
GCSE Simultaneous Equations Revision Guide
Find values that satisfy two equations at the same time.
Foundation
Exam practice
Worked examples
What You'll Learn
Understand a Simultaneous Solution
Recognise that the same pair of values must satisfy both equations.
Solve by Elimination
Match coefficients, add or subtract, then substitute back.
Solve by Substitution
Make one variable the subject and replace it in the other equation.
Solve Graphically
Read the intersection coordinate with appropriate accuracy.
Model Word Problems
Define variables and translate information into two equations.
Handle Non-linear Systems
Find all valid intersections of a line and a quadratic curve.
Key Rules and Methods
Scatter Graphs Lessons
Elimination Method
Add, subtract or scale equations to eliminate one variable efficiently.
Substitution Method
Replace one variable with an equivalent expression and solve the resulting equation.
CORE SUBTOPICS
Choosing a Method
Use elimination for convenient coefficients and substitution for an isolated variable.
Negative Coefficients
Track signs carefully when adding, subtracting and substituting.
Fractional or Decimal Solutions
Keep exact fractions where possible and convert only if requested.
Rearranging First
Make a variable the subject without changing equation equivalence.
Word Problems
Define unknown quantities with units before forming equations.
Linear-Quadratic Systems
Substitute the linear relation into the quadratic and retain every valid root.
Worked Example
Elimination When Both Equations Need Scaling
QUESTION
Solve 2x + 3y = 13 and 3x - 2y = 0.
- Given information: No coefficients are initially equal or opposite.
- Method: Create opposite y-coefficients, add the equivalent equations, solve for x, then substitute back.
- Multiply 2x + 3y = 13 by 2: 4x + 6y = 26.
- Multiply 3x - 2y = 0 by 3: 9x - 6y = 0.
- Add the new equations: 13x = 26.
- Divide by 13: x = 2.
- Substitute into 3x - 2y = 0: 6 - 2y = 0.
- Solve: y = 3.
- Final answer: x = 2 and y = 3.
- Independent answer check: 2(2) + 3(3) = 13 and 3(2) - 2(3) = 0, so both equations are satisfied.
- One common mistake: When scaling an equation, multiply every term, including the number on the right-hand side.
- Common GCSE Mistakes
- Combining Unaligned Terms
- Choosing the Wrong Operation
- Multiplying Only One Term
- Losing a Minus Sign
- Stopping After One Variable
- Substituting into the Same Rearranged Equation
- Ignoring a Second Non-linear Root
- Skipping the Check
- Exam Tips
- Number the equations so later references are clear.
- Align like terms and equality signs before elimination.
- Choose coefficients with the smallest convenient common multiple.
- Use brackets around a substituted expression.
- Keep exact fractions unless a decimal accuracy is requested.
- State what each variable represents in a word problem.
- Read graphical intersections using the scale and required accuracy.
- Substitute the final pair into both original equations.
Ready to Practice?
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 Simultaneous Equations Tutorial
Clear, step-by-step explanation of expanding brackets, simplifying complex expressions, and solving for x. Perfect for visual learners who need a breakdown of the rules in action.
- 12 minute comprehensive guide
- High-definition whiteboard walk-through
Continue Your GCSE Maths Revision
Continue Your GCSE Maths Revision
Frequently Asked Questions
They are equations solved together to find values that satisfy all equations at the same time.
Use it when one pair of coefficients is equal, opposite or easy to make equal.
Use it when a variable is already isolated or can be made the subject easily.
Brackets preserve the sign and operation applying to the entire substituted expression.
Brackets preserve the sign and operation applying to the entire substituted expression.
Replace the variables in both original equations and verify both equalities.
It is the coordinate of an intersection shared by both graphs.
Yes. Distinct parallel lines have no common point; coincident lines have infinitely many solutions.
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