Simultaneous Equations

Revise Simultaneous Equations for Additional Mathematics 0606 (O Level) — revision notes and instant AI marking.

📖 Revision notes · preview

Cambridge (CIE) IGCSE · Additional Maths

Simultaneous Equations

The big idea: when you've got two unknowns hiding in two clues, you combine the clues — by eliminating one letter or substituting one equation into the other — until only one unknown is left standing.

Summary — everything in this chapter, at a glance

Linear simultaneous equations only have x and y to the power 1 (no x², y², or xy). Solve them with the elimination method or the substitution method.
Quadratic (non-linear) simultaneous equations have an x², y², or xy in at least one equation. These must be solved by substitution — you always substitute the linear equation into the quadratic one, never the other way round.
Graphically, solving simultaneous equations = finding where two graphs cross. The coordinates of the crossing point(s) are the solution(s).
Quadratic simultaneous equations can have 0, 1, or 2 solutions — because a line can miss a curve, touch it (tangent), or cut through it twice.
Worded problems are just simultaneous equations wearing a costume — the hard part is translating the words into two equations correctly.
Always check your final answers in both original equations. This catches almost every sign error.

1. What Are Simultaneous Equations?

Two unknowns need two clues

Imagine someone tells you: "I'm thinking of two numbers. Their sum is 11." You genuinely can't figure out what those numbers are — there are infinite pairs that sum to 11 (1 and 10, 2 and 9, 5.5 and 5.5...). One clue for two unknowns just isn't enough information.

But now they tell you a second clue: "...and their difference is 3." Suddenly you can pin down exactly one pair of numbers. That's the whole idea behind simultaneous equations: when you have two unknowns, you need two equations (two independent pieces of information) to find unique values for both.

"Simultaneous" literally means "at the same time" — you're looking for the pair of values (an x and a y) that make both equations true simultaneously, not just one of them.

Example pair 3x + 2y = 11 and 2x − y = 5

Solution: x = 3, y = 1 — and this is the only pair that satisfies both equations at once.

Linear vs. Quadratic — how to tell them apart instantly

Look at the powers of x and y in each equation:

Type	What you'll see	Example
Linear	Only `x` and `y` — no squares, no `xy`	`3x + 2y = 11`
Quadratic / Non-linear	An `x²`, `y²`, or an `xy` term somewhere	`x² + y² = 25`

If you graph a linear equation, you get a straight line (linear = line — easy way to remember it). If you graph a quadratic equation like x² + y² = 25 you get a curve (in that case, a circle). This matters because two straight lines cross at most once, but a line and a curve can cross 0, 1, or 2 times — which is exactly why quadratic simultaneous equations can give you two different answer-pairs.

2. Linear Equations — The Elimination Method

Make one unknown disappear, solve for the other, then backtrack.

The core idea: cancel one letter out

6x x 6x − 6x = 0

The elimination recipe x y same sign subtract different signs add original

Worked example — start to finish

Practice Question 1

🔓 Read the full Simultaneous Equations note → You're seeing the preview · sign in to read it all

Also in the full note

Swap a letter for an expression
The solution is where lines cross
Turning a story into two equations
Why elimination won't work here

What's inside

📖 Revision notes 🎯 Learn mode ✦ AI flashcards ✓ Instant AI marking 🧊 3D explorers 🧪 Experiments & simulations 📈 Progress tracking

📄 Practise Simultaneous Equations with Additional Mathematics 0606 past papers Every paper with its mark scheme — answer online, marked instantly. Open →

Read the full Simultaneous Equations notes free

That's the preview — create a free account to read the rest, plus flashcards and practice questions with instant AI marking. No credit card.

Unlock the full notes free →

Simultaneous Equations