Where two conditions meet.

A simultaneous solution must satisfy both equations. Substitution replaces one variable using another equation. Elimination adds or subtracts suitable multiples so one variable cancels.

On a graph, the solution is the intersection of the two lines. Parallel distinct lines have no solution; equations describing the same line have infinitely many. Always substitute the proposed pair back into both original conditions.

INTERACTIVE MODEL

Try it. Watch it change.

y = 1x + 0y = −x + 2

Vertical scale / gradient a1 Intercept b0 Input x1

f(1) = 1

The graph is calculated from the displayed rule. Drag the highlighted point or use the sliders.

Explore: Turn on the comparison line y = −x + 2. Adjust your line until the intersection lies on the y-axis.

A common solution

\text{Both equations hold at the same }(x,y).

WORKED EXAMPLE

Solve x + y = 9 and $2x$ − y = 3.

Add the equations: 3x = 12, so x = 4.
Substitute: y = 9 − 4 = 5.
Check 2(4) − 5 = 3.

Assumed knowledge

Rearranging equations and handling negative signs.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Solve two linear equations together.
Interpret their intersection in a practical problem.

Year 10 preparation ↗

READY TO TRY IT?

Where two conditions meet.

Solve x + y = 9 and 2x2x2x − y = 3.

Put the idea to work.

Solve x + y = 9 and $2x$ − y = 3.