Solving Simultaneous Equations Graphically

What you'll learn

How to read coordinates from a graph accurately.
Why the point where two lines meet solves both equations.
How to draw straight-line graphs from equations.
How to use the intersection point to solve simultaneous equations.

Coordinates: the basics

A graph has two axes:

the horizontal axis is the x-axis
the vertical axis is the y-axis

A point is written as a coordinate pair.

Definition

Coordinate pair

A coordinate pair $(x, y)$ tells you the position of a point. The first number is the x-coordinate, and the second number is the y-coordinate.

Example

Reading a coordinate

Start at the origin, which is the centre point where the axes cross.
Move 4 squares to the right, so the x-coordinate is 4.
Move 1 square up, so the y-coordinate is 1.
The coordinate of the point is $(4, 1)$ .

The point (4, 1) is found by moving 4 squares right from the origin and then 1 square up.

Common Mistake

Writing coordinates backwards

Coordinates are always written x first, then y. So $(4, 1)$ is not the same as $(1, 4)$ .

What simultaneous equations mean

Simultaneous equations are equations that are solved at the same time.

Definition

Simultaneous equations

A pair of simultaneous equations has one solution if there is one pair of values for $x$ and $y$ that makes both equations true.

On a graph, each straight line shows all the points that satisfy one equation. The point where two lines meet satisfies both equations.

Two line graphs intersect at one point, which is the shared solution to both equations.

Key Idea

The key idea

The solution to two simultaneous equations is the coordinate of the point where the two graphs intersect.

Example

Checking a shared solution

For the equations $y = 2 x + 1$ and $x + y = 4$ , check whether $(1, 3)$ is a solution.

Substitute $x = 1$ and $y = 3$ into the first equation.
This gives $\times 1 + 1$ , which is true.
Substitute $x = 1$ and $y = 3$ into the second equation.
This gives $1 + 3 = 4$ , which is true.
So $(1, 3)$ is a solution to both equations.

The point (1, 3) lies on both y = 2x + 1 and x + y = 4, so it is their shared solution.

Reading the solution from two drawn graphs

Sometimes the lines are already drawn for you. Your job is to find where they cross.

To solve graphically:

Find the point where the two lines intersect.
Read the x-coordinate.
Read the y-coordinate.
Write the solution as $\text{...}$ and $\text{...}$ .

Example

Using two drawn lines

The graphs of $y = x + 1$ and $2 x + y = 7$ are drawn on the same grid. They cross at $(2, 3)$ .

Reading the intersection of the two drawn lines gives the solution x = 2, y = 3.

Find the point where the two straight lines meet.
Read across or down to the x-axis. The x-coordinate is 2.
Read across to the y-axis. The y-coordinate is 3.
Therefore the solution is $x = 2$ and $y = 3$ .

Drawing a straight-line graph

A straight-line graph is made by plotting points that satisfy an equation, then joining them with a ruler.

For equations like $y = 2 x - 1$ , you can choose some x-values, calculate the matching y-values, then plot the points.

Example

Drawing a line from y = 2x - 1

Draw the graph of $y = 2 x - 1$ .

Plotting points such as (0, -1), (1, 1) and (2, 3) lets you draw the straight line y = 2x - 1.

Choose some simple x-values, such as 0, 1 and 2.
Calculate the y-values.
$x=0⇒y=−1x=1⇒y=1x=2⇒y=3\begin{aligned} x = 0 &\Rightarrow y = -1 \\ x = 1 &\Rightarrow y = 1 \\ x = 2 &\Rightarrow y = 3 \end{aligned}$
Plot the points $(0, - 1)$ , $(1, 1)$ and $(2, 3)$ .
Join the points with a straight line using a ruler.

Tip

Use at least two points

Two points are enough to draw a straight line, but plotting three points helps you spot mistakes.

Rearranging before drawing

Some equations are not already written as $\text{...}$ . You may need to rearrange them first.

To make $y$ the subject means to get $y$ on its own on one side of the equation.

Example

Rearranging before drawing

Draw the graph of $2 y - x = 6$ .

$After rearranging to y = \frac{1}{2}x + 3, the points (0, 3), (2, 4) and (4, 5) lie on the line.$

Add $x$ to both sides to get $2 y = x + 6$ .
Divide both sides by 2 to get $\frac{1}{2}x + 3$ .
Choose easy x-values. If $x = 0$ , then $y = 3$ .
If $x = 2$ , then $y = 4$ .
If $x = 4$ , then $y = 5$ .
Plot $(0, 3)$ , $(2, 4)$ and $(4, 5)$ , then join them with a straight line.

Tip

Choose friendly values

If the equation has a fraction like $12x\frac{1}{2}x$ , choosing even x-values often gives whole-number y-values.

Solving by drawing both lines

If you are given two equations and no graph, draw both lines on the same set of axes. The answer is where they meet.

Example

Solving graphically

Solve the simultaneous equations $y = - x + 4$ and $2 y - x = 2$ using graphs.

Drawing both lines on the same axes shows that they meet at (2, 2).

Draw the first line, $y = - x + 4$ . Two easy points are $(0, 4)$ and $(4, 0)$ .
Rearrange the second equation: $2 y - x = 2$ becomes $\frac{1}{2}x + 1$ .
Draw the second line using points such as $(0, 1)$ , $(2, 2)$ and $(4, 3)$ .
Look carefully at the point where the two lines intersect.
The lines meet at $(2, 2)$ .
Therefore the solution is $x = 2$ and $y = 2$ .

Negative coordinates

Sometimes the intersection is to the left of the y-axis or below the x-axis. That means one or both coordinates are negative.

Example

Reading a negative intersection

The graphs of $y = x - 1$ and $y = - 2 x - 4$ are drawn on the same grid.

An intersection left of the y-axis and below the x-axis has negative coordinates, here (-1, -2).

Find the point where the lines intersect.
The crossing is one square left of the y-axis, so $x = - 1$ .
The crossing is two squares below the x-axis, so $y = - 2$ .
The solution is $x = - 1$ and $y = - 2$ .

Common Mistake

Parallel lines

If two straight lines are parallel, they never meet. That means there is no solution to the simultaneous equations.

Accuracy matters

Graphical solutions depend on how carefully you draw and read the graph. If the intersection is exactly on a grid point, give exact coordinates. If it is between grid lines, give the best estimate you can.

Exam technique

In the exam

Use a sharp pencil and a ruler when drawing straight lines.
Label each line or make it clear which equation it represents.
Read the intersection carefully: x-coordinate first, then y-coordinate.
If possible, substitute your answer back into the equations to check it makes sense.

Self review

Check yourself

Why does the intersection point solve both equations?
How do you know which number is the x-coordinate?
What should you do if the equation is not written in the form $\text{...}$ ?

Solving Simultaneous Equations Graphically

What you'll learn

How to read coordinates from a graph accurately.
Why the point where two lines meet solves both equations.
How to draw straight-line graphs from equations.
How to use the intersection point to solve simultaneous equations.

Coordinates: the basics

A graph has two axes:

the horizontal axis is the x-axis
the vertical axis is the y-axis

A point is written as a coordinate pair.

Definition

Coordinate pair

A coordinate pair $(x, y)$ tells you the position of a point. The first number is the x-coordinate, and the second number is the y-coordinate.

Example

Reading a coordinate

Start at the origin, which is the centre point where the axes cross.
Move 4 squares to the right, so the x-coordinate is 4.
Move 1 square up, so the y-coordinate is 1.
The coordinate of the point is $(4, 1)$ .

The point (4, 1) is found by moving 4 squares right from the origin and then 1 square up.

Common Mistake

Writing coordinates backwards

Coordinates are always written x first, then y. So $(4, 1)$ is not the same as $(1, 4)$ .

What simultaneous equations mean

Simultaneous equations are equations that are solved at the same time.

Definition

Simultaneous equations

A pair of simultaneous equations has one solution if there is one pair of values for $x$ and $y$ that makes both equations true.

On a graph, each straight line shows all the points that satisfy one equation. The point where two lines meet satisfies both equations.

Two line graphs intersect at one point, which is the shared solution to both equations.

Key Idea

The key idea

The solution to two simultaneous equations is the coordinate of the point where the two graphs intersect.

Example

Checking a shared solution

For the equations $y = 2 x + 1$ and $x + y = 4$ , check whether $(1, 3)$ is a solution.

Substitute $x = 1$ and $y = 3$ into the first equation.
This gives $\times 1 + 1$ , which is true.
Substitute $x = 1$ and $y = 3$ into the second equation.
This gives $1 + 3 = 4$ , which is true.
So $(1, 3)$ is a solution to both equations.

The point (1, 3) lies on both y = 2x + 1 and x + y = 4, so it is their shared solution.

Reading the solution from two drawn graphs

Sometimes the lines are already drawn for you. Your job is to find where they cross.

To solve graphically:

Find the point where the two lines intersect.
Read the x-coordinate.
Read the y-coordinate.
Write the solution as $\text{...}$ and $\text{...}$ .

Example

Using two drawn lines

The graphs of $y = x + 1$ and $2 x + y = 7$ are drawn on the same grid. They cross at $(2, 3)$ .

Reading the intersection of the two drawn lines gives the solution x = 2, y = 3.

Find the point where the two straight lines meet.
Read across or down to the x-axis. The x-coordinate is 2.
Read across to the y-axis. The y-coordinate is 3.
Therefore the solution is $x = 2$ and $y = 3$ .

Drawing a straight-line graph

A straight-line graph is made by plotting points that satisfy an equation, then joining them with a ruler.

For equations like $y = 2 x - 1$ , you can choose some x-values, calculate the matching y-values, then plot the points.

Example

Drawing a line from y = 2x - 1

Draw the graph of $y = 2 x - 1$ .

Plotting points such as (0, -1), (1, 1) and (2, 3) lets you draw the straight line y = 2x - 1.

Choose some simple x-values, such as 0, 1 and 2.
Calculate the y-values.
$x=0⇒y=−1x=1⇒y=1x=2⇒y=3\begin{aligned} x = 0 &\Rightarrow y = -1 \\ x = 1 &\Rightarrow y = 1 \\ x = 2 &\Rightarrow y = 3 \end{aligned}$
Plot the points $(0, - 1)$ , $(1, 1)$ and $(2, 3)$ .
Join the points with a straight line using a ruler.

Tip

Use at least two points

Two points are enough to draw a straight line, but plotting three points helps you spot mistakes.

Rearranging before drawing

Some equations are not already written as $\text{...}$ . You may need to rearrange them first.

To make $y$ the subject means to get $y$ on its own on one side of the equation.

Example

Rearranging before drawing

Draw the graph of $2 y - x = 6$ .

$After rearranging to y = \frac{1}{2}x + 3, the points (0, 3), (2, 4) and (4, 5) lie on the line.$

Add $x$ to both sides to get $2 y = x + 6$ .
Divide both sides by 2 to get $\frac{1}{2}x + 3$ .
Choose easy x-values. If $x = 0$ , then $y = 3$ .
If $x = 2$ , then $y = 4$ .
If $x = 4$ , then $y = 5$ .
Plot $(0, 3)$ , $(2, 4)$ and $(4, 5)$ , then join them with a straight line.

Tip

Choose friendly values

If the equation has a fraction like $12x\frac{1}{2}x$ , choosing even x-values often gives whole-number y-values.

Solving by drawing both lines

If you are given two equations and no graph, draw both lines on the same set of axes. The answer is where they meet.

Example

Solving graphically

Solve the simultaneous equations $y = - x + 4$ and $2 y - x = 2$ using graphs.

Drawing both lines on the same axes shows that they meet at (2, 2).

Draw the first line, $y = - x + 4$ . Two easy points are $(0, 4)$ and $(4, 0)$ .
Rearrange the second equation: $2 y - x = 2$ becomes $\frac{1}{2}x + 1$ .
Draw the second line using points such as $(0, 1)$ , $(2, 2)$ and $(4, 3)$ .
Look carefully at the point where the two lines intersect.
The lines meet at $(2, 2)$ .
Therefore the solution is $x = 2$ and $y = 2$ .

Negative coordinates

Sometimes the intersection is to the left of the y-axis or below the x-axis. That means one or both coordinates are negative.

Example

Reading a negative intersection

The graphs of $y = x - 1$ and $y = - 2 x - 4$ are drawn on the same grid.

An intersection left of the y-axis and below the x-axis has negative coordinates, here (-1, -2).

Find the point where the lines intersect.
The crossing is one square left of the y-axis, so $x = - 1$ .
The crossing is two squares below the x-axis, so $y = - 2$ .
The solution is $x = - 1$ and $y = - 2$ .

Common Mistake

Parallel lines

If two straight lines are parallel, they never meet. That means there is no solution to the simultaneous equations.

Accuracy matters

Exam technique

In the exam

Use a sharp pencil and a ruler when drawing straight lines.
Label each line or make it clear which equation it represents.
Read the intersection carefully: x-coordinate first, then y-coordinate.
If possible, substitute your answer back into the equations to check it makes sense.

Self review

Check yourself

Why does the intersection point solve both equations?
How do you know which number is the x-coordinate?
What should you do if the equation is not written in the form $\text{...}$ ?

Solving Simultaneous Equations Graphically

What you'll learn

Coordinates: the basics

What simultaneous equations mean

Reading the solution from two drawn graphs

Drawing a straight-line graph

Rearranging before drawing

Solving by drawing both lines

Negative coordinates

Accuracy matters

Recap questions

You've reached the end

Solving Simultaneous Equations Graphically

What you'll learn

Coordinates: the basics

What simultaneous equations mean

Reading the solution from two drawn graphs

Drawing a straight-line graph

Rearranging before drawing

Solving by drawing both lines

Negative coordinates

Accuracy matters

Recap questions

You've reached the end

Solving Simultaneous Equations Graphically

What you'll learn

Coordinates: the basics

What simultaneous equations mean

Reading the solution from two drawn graphs

Drawing a straight-line graph

Rearranging before drawing

Solving by drawing both lines

Negative coordinates

Accuracy matters

Recap questions

You've reached the end

Solving Simultaneous Equations Graphically Revision Guide

Solving Simultaneous Equations Graphically

What you'll learn

Coordinates: the basics

What simultaneous equations mean

Reading the solution from two drawn graphs

Drawing a straight-line graph

Rearranging before drawing

Solving by drawing both lines

Negative coordinates

Accuracy matters

Recap questions

You've reached the end

Solving Simultaneous Equations Graphically Revision Guide