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