14.3 Solving simultaneous equations graphically — Learn

Prerequisites: Plot coordinates and draw straight-line graphs from tables or equations.

Objectives: Solve a pair of linear equations by finding the intersection of their graphs. This page covers A19.

The intersection is the solution

A point $(x,y)$ lies on a graph when its coordinates satisfy that graph's equation. The intersection of two graphs satisfies both equations, so it gives their simultaneous solution.

Example. Find the intersection of $y=2x+1$ and $y=-x+7$.

At the intersection, the two expressions for $y$ are equal:

Substitute $x=2$ into $y=2x+1$:

The lines meet at $(2,5)$. On a graph, read both coordinates from the crossing. A drawn graph may give an approximate answer; check by substitution when possible.

Guided investigation

Predict before moving the slider: at which integer $x$ do you think the two lines meet? Then test values and compare their $y$ coordinates.

Explain: What do equal $y$ values tell you about the point? Enter your explanation in your notes.

Watch out. The intersection is an ordered pair $(x,y)$, not one number. Read across for $x$ and up for $y$, then check that the pair fits both equations.

Progress check

Plot Company A’s fare points $(4,14)$ and $(8,22)$, then Company B’s $(4,16)$ and $(8,24)$. What do the two gradients tell you about the per-kilometre prices?

You should now be able to

• explain why an intersection solves both equations;
• find an exact intersection using algebra;
• estimate an intersection from a graph and check it.