Simultaneous Equations
One equation with two unknowns has endless solutions. Two equations, both true at once, pin the unknowns down to a single pair. On a graph that pair is where two lines cross; on paper it's found by making one unknown disappear.
National curriculum KS3: solve two simultaneous linear equations in two variables (algebraically and graphically); GCSE adds linear/quadratic pairs1 · Why one equation isn't enough
x + y = 10. What are x and y? Could be 3 and 7, or 1 and 9, or 12 and −2. Every point on a line satisfies it. Drag along the line: infinitely many answers. Now add a second fact: x − y = 4. Only one point sits on both lines. That's the solution, and the only one.
2 · Solving by drawing
Plot both lines (see the linear graphs page) and read the crossing point. It's quick and it shows you what the answer means, but reading a graph is only as accurate as your pencil. Set two equations and compare the graphical answer with the exact one.
3 · Elimination: the shopping-bill idea
Before any algebra, here's the whole method in a shop. Two apples and a banana cost 90p. Two apples and three bananas cost £1.50. The apples are the same in both bills, so the difference between the bills must be due to the bananas alone: two extra bananas cost 60p, so one banana is 30p. Subtracting one bill from the other eliminated the apples.
That's elimination. Line the equations up, make the coefficients of one unknown match, then add or subtract so that unknown vanishes. Solve the one-unknown equation that's left, then substitute back to find the other.
4 · Elimination step by step
Choose an equation pair. The solver shows each move; click through so you see why each is made. Pairs are grouped by difficulty: matching coefficients first, then one equation needs multiplying, then both.
Same signs subtract, different signs add. If the matched terms are +3y and +3y, subtracting removes them. If they're +3y and −3y, adding does. Students who always subtract get stuck on the second kind; students who always add get stuck on the first. Look at the signs, then choose. And when you subtract, subtract every term, including the right-hand side.
5 · Substitution: the other route
If one equation already has a lonely letter, like y = 2x − 1, there's a shortcut. You know what y equals, so put that expression in place of y in the other equation. Now there's only x. This is the method you'll need at GCSE when one equation is a curve.
6 · From words to equations
Exam questions rarely hand you the equations. They describe a situation with two unknowns and two facts. The skill is translating. Each fact becomes one equation; the unknowns are whatever the question asks for.
7 · When it goes wrong (and what that means)
| What happens | Example | On the graph | Meaning |
|---|---|---|---|
| Both unknowns vanish, leaving something false | x + y = 5 and x + y = 8 → 0 = −3 | Parallel lines | No solution. The facts contradict each other. |
| Both vanish, leaving something true | x + y = 5 and 2x + 2y = 10 → 0 = 0 | The same line twice | Infinitely many solutions. The second fact is the first in disguise. |
| Fractions appear | x = 7/3 | Lines cross between grid points | Perfectly fine. Keep it as a fraction; don't round. |
8 · Where simultaneous equations show up
Pricing
Two orders, two totals, unknown unit prices: exactly the shopping-bill example. Businesses back out component prices from bundled invoices this way.
Mixing
How much 20% solution and 50% solution to mix for 10 litres of 35%? Volume equation plus concentration equation, two unknowns. Chemistry, cooking, paint.
Break-even and tariffs
Two phone plans, two cost formulas: where they cross is the usage at which they cost the same. Below it one is cheaper, above it the other.
Computing and graphics
Where does a ray hit a surface? Where do two roads on a map intersect? Every line-crossing in a game engine is a pair of simultaneous equations solved millions of times a second.
Science
Two balanced forces, two circuit loops (Kirchhoff's laws), two reactions sharing a reagent: nature keeps handing you two facts about two unknowns.
Bigger systems
Three unknowns need three equations; a weather model has millions. The method scales (it becomes matrix algebra), but the idea is still "eliminate until one unknown is left".
Worked examples
Solve 3x + 2y = 16 and 5x + 2y = 24
Solve 2x + 3y = 13 and x + 2y = 8
Solve y = 2x − 1 and 3x + y = 11 by substitution
Exam-style: 3 coffees and 2 teas cost £8.70. 2 coffees and 5 teas cost £11.30. Find the price of each.
Quick check
What you've learned
Two unknowns need two independent facts. Graphically the solution is where two lines cross. Algebraically you eliminate one unknown by matching its coefficient in both equations (multiplying if needed) and then adding when the signs differ or subtracting when they match; or you substitute an expression for one unknown into the other equation. Either way you finish by substituting back and checking in both originals. Parallel lines mean no solution; identical lines mean infinitely many. In word problems, name the unknowns first, then turn each fact into an equation.