Two Unknowns, One Answer: Solving Simultaneous Equations

Imagine you buy 2 apples and 1 banana for £5, and on another day you buy 2 apples and 3 bananas for £9. You have two clues and two unknowns — and that is exactly enough to find both prices. How? That question drives today's whole lesson. Before we dive in, try this warm-up: if x = 3 and y = 2, what is the value of x + y, and what is the value of 2x − y? Jot down your answers.

A simultaneous equation is one equation that contains two unknowns, such as x and y. One equation on its own has infinitely many solutions — any pair (x, y) that satisfies it. But when you have two equations that must both be true at the same time (simultaneously), there is usually exactly one pair of values that satisfies them both. That pair is the solution. We write a pair like this: Equation 1: x + y = 7 Equation 2: x − y = 3 The notation 'x + y = 7' means: whatever values x and y take, they must add up to 7. Both equations must hold at once. **Method: Elimination** The elimination method removes one unknown by adding or subtracting the two equations. Step 1 — Look at the equations. If the same unknown has the same coefficient (the number in front of it) in both equations, you can eliminate it. x + y = 7 …(1) x − y = 3 …(2) Both equations have '1x'. Subtract equation (2) from equation (1): (x + y) − (x − y) = 7 − 3 2y = 4 y = 2 Step 2 — Substitute y = 2 back into equation (1) to find x: x + 2 = 7 x = 5 Step 3 — Check in equation (2): 5 − 2 = 3 ✓ The solution is x = 5, y = 2. This means there is exactly one point where both equations are satisfied. **What if coefficients don't match?** Sometimes you must multiply one (or both) equations first to make a coefficient match. For example, if one equation has '2y' and another has 'y', multiply the second by 2 before eliminating. You will practise this in the questions below.