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Knowledge Organiser: Simultaneous Equations

Part of Simultaneous Equations · GCSE GCSE Mathematics revision

This topic summary covers Knowledge Organiser: Simultaneous Equations within Simultaneous Equations for GCSE Mathematics. Revise Simultaneous Equations in Algebra for GCSE Mathematics with 15 exam-style questions and 12 flashcards. This is a high-frequency topic, so it is worth revising until the explanation feels precise and repeatable. It is section 8 of 8 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.

Topic position

Section 8 of 8

Practice

15 questions

Recall

12 flashcards

Knowledge Organiser: Simultaneous Equations

Key Terms
  • Simultaneous equations: Two equations with two unknowns solved at the same time
  • Elimination: Adding or subtracting equations to remove one variable
  • Substitution: Expressing one variable in terms of the other, then substituting
  • Solution: The pair of values (x, y) satisfying both equations
  • Linear-quadratic: One linear and one quadratic equation — can give 2 solutions
Must-Know Facts
  • Two linear equations have at most ONE solution (where the lines cross)
  • Elimination: make the coefficients of one variable the same, then add or subtract
  • Add equations when coefficients have OPPOSITE signs; subtract when SAME signs
  • After finding x (or y), substitute back to find the other variable
  • Linear + quadratic: rearrange linear to y = …, substitute into quadratic
  • Always check both values satisfy BOTH original equations
Key Methods
  • Elimination: multiply equations to match one coefficient, then add/subtract
  • Substitution: rearrange one equation, substitute into the other
  • Graphical: plot both lines — intersection = solution
  • Linear-quadratic: substitute the linear into the quadratic, solve, find both pairs
Common Mistakes
  • Subtracting when you should add: Add equations when the matching coefficients have opposite signs; subtract when they have the same sign — getting this wrong gives a wrong or unsimplified equation
  • Forgetting to find both variables: After eliminating one variable and solving, always substitute back to find the second variable
  • Not checking the solution: Substitute both x and y values into BOTH original equations to verify they both balance
  • Linear-quadratic: pairing solutions incorrectly: Match each x value with its corresponding y value — substituting x = 2 gives one y, and x = −1 gives a different y

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Read this section alongside the surrounding pages in Simultaneous Equations. That gives you the full topic sequence instead of a single isolated revision point.

Common Mistakes to AvoidTwo Equations, Two UnknownsTwo Methods

Practice Questions for Simultaneous Equations

Which method is most efficient for solving the following simultaneous equations? 5x + 2y = 14 3x + 2y = 10

  • A. Substitution — rearrange the first equation for x, then substitute into the second.
  • B. Elimination — subtract the equations directly because the y-coefficients are already equal.
  • C. Elimination — multiply both equations by 5 and 3 respectively to match the x-coefficients.
  • D. Trial and improvement — test integer values of x and y until both equations are satisfied.
1 markfoundation

Ali solves the simultaneous equations 5x + y = 17 and x - y = 1 by elimination. Ben says he can also use substitution by rearranging x - y = 1 to get x = y + 1. Explain how elimination works for these two equations and state one advantage of Ben's substitution approach.

3 marksstandard

Quick Recall Flashcards

Steps for the elimination method
1. Match one set of coefficients (multiply if needed) 2. Same signs: subtract the equations 3. Different signs: add the equations 4. Solve for the remaining variable 5. Substitute back to find the other variable
What are simultaneous equations?
Two or more equations that share the same variables. Solving them finds the values of those variables that satisfy ALL equations at the same time.

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