Graphical Solution of Simultaneous Equations

Solve two simultaneous equations by drawing both lines and reading the point where they cross.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1 · 6 real past-paper questions below

اقرأ هذا الدرس مشروحاً بالعربي ←

The Crossing Point Is the Answer

Each equation is a straight line. Every point on a line makes its equation true.

Rule

Only the point where the two lines cross is on both lines. Its x and y solve both equations at once.

Key idea

Two lines that are not parallel cross once, so there is exactly one answer.

Read the Crossing Point

The 1-mark question: the lines are already drawn. Write down the solution of the two equations.

Worked example

1
$x + 2 y = 8 and y = 2 x - 1$ — The two lines on the grid. The answer is the one point that is on both lines.
2
$they cross at (2, 3)$ — Find where the lines cross. Only that point makes both equations true.
3
$x = 2$ — Look straight down from the crossing to the x-axis. That number is x.
4
$y = 3$ — Look straight across to the y-axis. That number is y.
5
$2 + 2 (3) = 2 + 6 = 8$ — Check in the first equation: put x = 2 and y = 3 in. It gives 8, so it fits.
6
$2 (2) - 1 = 4 - 1 = 3$ — Check in the second equation too. It gives y = 3, so the point is on both lines.

Key idea

Match the answer to the words. “Write down the coordinates” → (2, 3). “Solve” → x = 2, y = 3. If one line is flat, like $y = 1$ , every point on it has y = 1, so you only need to read x.

Common mistake

Swapping them and writing $x = 3, y = 2$ . The first number of a point is always x.

Draw Both Lines First

Sometimes you must draw the lines yourself. A line like $a x + b y = c$ has x and y on the same side, so the quickest two points are where it meets each axis:

Formula

a x + b y = c meets the axes at (0, \frac{c}{b}) and (\frac{c}{a}, 0)

$a$ is the number in front of x, $b$ the number in front of y, and $c$ the number on the other side. Put x = 0 to get the first point, and y = 0 to get the second.

Worked example

1
$2 x + 3 y = 12$ — The first line. Here a = 2, b = 3, c = 12.
2
$x = 0 : 2 (0) + 3 y = 12$ — Put x = 0, because every point on the y-axis has x = 0. The 2x term becomes 0.
3
$3 y = 12, y = 4$ — Divide both sides by 3. The line meets the y-axis at (0, 4).
4
$y = 0 : 2 x + 3 (0) = 12$ — Now put y = 0, because every point on the x-axis has y = 0.
5
$2 x = 12, x = 6$ — Divide both sides by 2. The line meets the x-axis at (6, 0). Join (0, 4) to (6, 0) with a ruler.
6
$y = x - 1 : x = 0 \Rightarrow y = - 1$ — The second line is already “y = …”, so pick an easy x. x = 0 gives y = −1, the point (0, −1).
7
$x = 4 \Rightarrow y = 4 - 1 = 3$ — Pick a second x far from the first, so the ruler line is accurate. This gives (4, 3). Join the two points.
8
$cross at (3, 2)$ — Read the crossing: down to x = 3, across to y = 2.
9
$2 (3) + 3 (2) = 12, 3 - 1 = 2$ — Check in both equations. Both are true, so x = 3, y = 2.

Common mistake

Mixing up the two axis points: (0, 4) is on the y-axis, (6, 0) is on the x-axis. Check each one fits $2 x + 3 y = 12$ .

When Lines Cross Between Grid Lines

Lines do not always cross on a grid corner. Then your answer is an estimate, read to the nearest small square.

Worked example

1
$x + y = 5 and y = 2 x + 1$ — The two lines. They cross between grid lines, so the answer will not be whole numbers.
2
$1 < x < 1.5$ — Look straight down from the crossing. It lands a little past 1, before 1.5 on the x-axis.
3
$x \approx 1.3$ — Count the small squares past 1 to read it to 1 decimal place.
4
$y \approx 3.7$ — Look straight across to the y-axis and read y the same way.
5
$1.3 + 3.7 = 5$ — Check in the first equation. It gives 5, so it fits.
6
$2 (1.3) + 1 = 3.6 \approx 3.7$ — Check in the second. 3.6 is very close to 3.7, which is what an estimate should give.
7
$x \approx 1.3, y \approx 3.7$ — The answer, to 1 decimal place. Algebra gives the exact point, about (1.33, 3.67).

Key idea

The mark scheme accepts a small range around the true crossing. If your own line is a little off, you still get the mark for reading where your lines cross, so always write an answer.

Common mistake

Rounding to whole numbers, like (1, 4). It must fit both equations. 1 + 4 = 5 works, but 2(1) + 1 = 3, not 4.

Which Method? Read the Question

Read how the question wants the answer.

You see	You do
Lines already drawn	read the crossing point
“Draw” a line with x and y together	find where it meets each axis
“Draw” a line given as y = …	Drawing Straight-Line Graphs
“Solve”, no graph	Simultaneous Equations by Elimination
The same line twice	infinitely many solutions
Two parallel lines	no solution
A line and a curve	Solving Equations by Graphical Methods

Key idea

A graph answer can be slightly off when the lines cross between grid lines. Algebra gives the exact answer.

Practise graphical solution of simultaneous equations now

Interactive visual aids, then real past-paper questions marked on your working like the exam. Free to start.

Practise free on Lemmara

Real past-paper questions on graphical solution of simultaneous equations

Each with a full worked solution.

Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education or Pearson Edexcel. The lessons and worked examples are Lemmara’s own.