Cambridge IGCSE · International Maths (Extended)
Simultaneous Equations
The big idea: when you have two unknowns, you need two equations —
and "solving simultaneously" just means finding the one pair of values that makes
both equations true at the same time.
📋 Summary — What's in this chapter
- Linear simultaneous equations — two straight-line equations (no
x², y², or xy terms), solved by elimination, substitution, or graphically.
- Elimination method — match the coefficients of one letter, then add or subtract the equations to cancel it out.
- Substitution method — rearrange one equation to isolate a letter, then plug it into the other equation.
- Graphical method — plot both lines; the solution is their point of intersection.
- Forming your own equations from worded problems (e.g. cost of items, sums and differences of numbers).
- Quadratic (non-linear) simultaneous equations — one equation has an
x², y², or xy term. Always solved by substitution — never elimination.
- Quadratic simultaneous equations can give two solutions, one solution (tangent), or no solutions at all.
1 · What Are Simultaneous Equations?
Imagine I tell you: "I'm thinking of two numbers. Their sum is 19." Can you tell me exactly
what the two numbers are? No — there are infinite possibilities (10 and 9, 15 and 4, 1 and 18...).
One equation with two unknowns has endless solutions.
But now I add a second clue: "...and their difference is 5." Suddenly, there's only
one pair of numbers that satisfies both clues at once: 12 and 7. That's
the whole idea of simultaneous equations — two unknowns need two equations to pin
down a single, unique answer (in the linear case).
Think of it like this
Each equation is a "rule" that a huge family of (x, y) pairs could satisfy. Solving simultaneously means finding the one pair (or pairs) that obeys every rule at once.
There are two types you'll meet in this chapter:
- Linear simultaneous equations — no squared terms or
xy terms. Example: 3x + 2y = 11 and 2x − y = 5.
- Quadratic (non-linear) simultaneous equations — at least one equation contains
x², y², or xy. Example: x² + y² = 25 and y − 2x = 5.
2 · Solving Linear Equations by Elimination
Elimination means making one of the letters disappear by combining the two equations
— either by adding them together or subtracting one from the other.
The Method, Step by Step
- Number your equations ① and ② — this keeps your working organised and stops you losing track.
- Match the coefficients of x (or y) in both equations. Multiply each entire equation by whatever number is needed. Whatever you do to one term, you must do to every term in that equation.
- Add or subtract the equations to eliminate that letter (use the golden rule above).
- Solve the resulting one-letter equation.
- Substitute that value back into either original equation to find the other letter.
- Check both values in the other original equation — this is your safety net against arithmetic slips.
Worked Example
5x + 2y = 11
4x − 3y = 18
1.
2.