Simultaneous Equations

Simultaneous equations are pairs of equations that share the same unknowns. You need to find values of the unknowns that satisfy both equations at the same time ("simultaneously"). This topic is a key part of the AQA GCSE Mathematics specification and appears regularly on both Foundation and Higher tier papers.

---

Why Simultaneous Equations?

A single equation with two unknowns (e.g. x + y = 10) has infinitely many solutions. To find a unique solution, you need a second equation. Together, the two equations give one pair of values (or occasionally more) that satisfy both.

---

Method 1: Elimination

The elimination method works by adding or subtracting the two equations to eliminate one of the variables.

Steps

1. Label the equations (1) and (2).
2. Make the coefficients of one variable the same in both equations (by multiplying one or both equations).
3. Add or subtract the equations to eliminate that variable.
4. Solve for the remaining variable.
5. Substitute back to find the other variable.

Worked Example 1

Solve:

Equation (1): 2x + y = 7

Equation (2): x + y = 4