Elimination or substitution: which is better?

Elimination is usually quicker for two linear equations. Substitution is the only reliable choice when one equation is quadratic.

What does the solution mean on a graph?

It is where the two graphs cross. Two straight lines cross once; a line and a curve can cross twice, which is why that case gives two pairs of answers.

How to Solve Simultaneous Equations (3 Methods)

Short answer

How do you solve simultaneous equations?

Use elimination when the equations line up: make one variable's coefficients match, then add or subtract to remove it. Use substitution when one equation gives a variable directly, or when one equation is quadratic. Always check the answer in both equations.

Two equations, two unknowns: you want the one pair of values that makes both equations true. On a graph, that is where the lines cross.

Try it: where do the two lines cross?

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The lines $y = x + 1$ and $y = - x + 5$ cross at $(2, 3)$ , the only point on both.

Method 1: elimination

Solve $2 x + 3 y = 12$ and $x - y = 1$ .

Multiply the second equation by 3 so the $y$ terms match:

3 x - 3 y = 3

Add it to the first equation. The $y$ terms cancel:

5 x = 15 ⟹ x = 3

Substitute back: $3 - y = 1 $, so$ y = 2$.

Check in the other equation: $2 (3) + 3 (2) = 12$ .

Method 2: substitution

The second equation gives $x = y + 1$ directly. Put it into the first:

2 (y + 1) + 3 y = 12 ⟹ 5 y = 10 ⟹ y = 2, x = 3

Same answer, different route.

When one equation is quadratic

Solve $y = x + 1$ and $x^{2} + y^{2} = 13$ . Substitute the line into the curve:

x^{2} + (x + 1)^{2} = 13 ⟹ 2 x^{2} + 2 x - 12 = 0 ⟹ x^{2} + x - 6 = 0

(x + 3) (x - 2) = 0 ⟹ x = 2 or x = - 3

Find each $y$ from the line: $x = 2$ gives $y = 3$ , and $x = - 3$ gives $y = - 2$ . The line crosses the circle twice, so there are two solution pairs.

Mistakes that cost marks

Subtracting when you should add. Signs that differ cancel by adding.
Finding $x$ and stopping. The answer is a pair.
Pairing the wrong $y$ with each $x$ in the quadratic case. Substitute each $x$ into the line separately.

How to solve simultaneous equations