Solving Simultaneous Equations Graphically

The idea in one sentence

Every point on a line makes that line’s equation true; so the point where two lines cross makes both equations true at once, and that is the solution.

The method

Rearrange each equation into the form $y = \dots$ if it is not there already.
Make a short table for each — three points is plenty for a straight line, and the third is a useful check on the other two.
Plot both lines on the same axes, with a sensible scale that includes where you expect them to meet.
Read off the crossing point. That is your $x$ and $y$ .
Substitute back into both original equations to confirm.

Why step 5 really matters

Graphical solutions are read off a picture. If the crossing point sits between grid lines, or your ruler wandered slightly, you will get an answer that is close but wrong.

Substituting back takes ten seconds and either confirms the answer or tells you to look again. If your values do not fit both equations exactly, the true solution is probably not at a whole-number point — and a graph may simply not be precise enough, which is one reason algebraic methods exist.

When the lines do not cross

Not every pair of equations has a solution:

Gradients	Intercepts	Picture	Solutions
Different	Anything	Lines cross once	Exactly one
Same	Different	Parallel lines	None
Same	Same	The same line twice	Infinitely many

You can spot all three cases before drawing anything, just by comparing $m$ and $c$ . If someone asks you to solve $y = 2x + 1$ and $y = 2x - 3$ , you can say “no solution — they are parallel” without touching a pencil.

Key rules and formulas

The point of intersection is the solution

A point on a line satisfies that line's equation. A point on both lines satisfies both equations — which is exactly what "solve simultaneously" asks for.

Rearrange first if you need to

Equations like

2 x + y = 8

are easier to plot once rearranged into

y = 8 - 2 x

. Get

y

on its own, then build a table.

Always check

Substitute your

x

and

y

into both original equations. Both must balance. Reading a graph involves estimating, so this step is not optional.

Parallel lines never meet

If the two equations have the same gradient but different intercepts, the lines are parallel and there is no solution. If they have the same gradient and the same intercept, they are the same line and there are infinitely many.

Worked examples

Worked example 1

Solving by drawing

Solve

y = x + 1

and

y = 5 - x

graphically.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Rearranging before plotting

Solve

2 x + y = 8

and

y = x - 1

graphically.

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

When there is no solution

What happens when you try to solve

y = 2 x + 1

and

y = 2 x - 3

graphically?

Try it yourself first, then reveal the solution one step at a time.

$x$	$0$	$2$	$4$
$y = 8 - 2 x$	Row 1, 0	Row 1, 2	Row 1, 4
$y = x - 1$	Row 2, 0	Row 2, 2	Row 2, 4

x

0

2

4

y = 8 - 2 x

y = x - 1

Solving Simultaneous Equations Graphically

In this lesson you will learn to…

The idea in one sentence

The method

Why step 5 really matters

When the lines do not cross

Key rules and formulas

The point of intersection is the solution

Rearrange first if you need to

Always check

Parallel lines never meet

Move a line to find a crossing point

Worked examples

Solving by drawing

Rearranging before plotting

When there is no solution

Practice: simultaneous equations from graphs