• 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.

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.

Linear vs. Quadratic — how to tell them apart instantly

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

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.

The core idea: cancel one letter out

Elimination works like this: if two equations both contain, say, 6x, and you subtract one equation from the other, the x's vanish completely (6x − 6x = 0). You're left with a single equation in a single unknown — which you can just solve directly.

Worked example — start to finish

Swap a letter for an expression

Substitution takes a different route to the same destination. Instead of cancelling a term, you rearrange one equation so that y (or x) is completely on its own — "the subject" of the equation. Then, wherever y appears in the other equation, you replace it with that expression. This turns two equations with two unknowns into one equation with just one unknown.

Worked example

The solution is where lines cross

Every equation in x and y can be drawn as a graph. A linear equation draws a straight line; every point on that line makes the equation true. If you draw two equations on the same axes, then any point where the two graphs intersect is a point that satisfies both equations at once — which is exactly what "solving simultaneously" means.

        y
        |        2x−y=3
        |          /
        |         /
        |    ★(2,1)  ← lines cross here
        |       /  \
        |      /    \
        |_____/______\____ x
        |    /        3x+y=7
      

Reading off the graph: the lines above cross at (2, 1), so the solution to 2x − y = 3 and 3x + y = 7 simultaneously is x = 2, y = 1.

Turning a story into two equations

Worded questions are simultaneous equations in disguise. The skill being tested isn't really the algebra — it's spotting that there are two unknown quantities and two independent pieces of information, and translating each piece of information into an equation. Always start by clearly defining your letters, in words, before writing any equations.

Worked example

Why elimination won't work here

Elimination relies on lining up matching terms and cancelling them by adding or subtracting. That's only possible when both equations have the same "shape" of terms. Once one equation has an x² or xy and the other doesn't, there's nothing to match up — so elimination is off the table. Substitution, however, always works because you're not trying to match anything; you're replacing a whole variable with an expression, however complicated that expression is.

Worked example — full method

Reading the number of solutions from the discriminant

Once you've substituted and simplified down to a quadratic equation (something you'd solve with factorising, completing the square, or the quadratic formula), the discriminant tells you geometrically what's happening between the line and the curve: