# How to Solve Simultaneous Equations Using the Graphical Method

The graphical method is used to solve simultaneous equations.

It involves plotting each of the simultaneous equations - which here are linear equations - as lines on a graph. (Have a go at drawing a graph from a linear equation.)

The x and y coordinates of the point where the lines intersect gives the solution to the simultaneous equations.

General Approach to Using the Graphical Method:

Step 1
Plot Equation (1) as a line on a pair of axes.

Step 2
Plot Equation (2) as a line on the same pair of axes.

Step 3
Find the point where the two lines cross.

Step 4
The x-coordinate of the point found in Step 3 gives the value of x that solves the simultaneous equations.

Step 5
The y-coordinate of the point found in Step 3 gives the value of y that solves the simultaneous equations.

# A Real Example of Solving Simultaneous Equations Using the Graphical Method

Question: What values of x and y solve the simultaneous equations below?

Step 1
Plot Equation (1) as a line on a pair of axes.

Equation (1) is a linear function. When plotted, it forms a line.

Rearrange Equation (1) put it into the standard form for the equation of a straight line.

Firstly, make y the subject.

This is an algebraic equation that contains addition.

To make y the subject, look at what is being done to the y:

The opposite of adding x is subtracting x, and this should be done to both sides:

y is now the subject of the equation.

To make this into the standard form for the equation of a straight line, swap the 5 and the -x around:

This is in the standard form for the equation of a line:

This allows the line to be plotted on a pair of axis.

Mark the y-intercept on the y-axis:

Then use the slope to draw the rest of the line. The slope is -1, so the line goes down one unit for every unit it goes across:

Equation (1) drawn on a pair of axes looks like:

Step 2
Plot Equation (2) as a line on the same pair of axes.

The same process for Step 1 could be followed for Step 2, but a different approach will be attempted here.

Find where Equation (2) crosses the y-axis by substituting x = 0 into the equation:

Rearranging and solving for y:

This is the y-intercept and can be plotted on the same pair of axes as Equation (1):

Find where Equation (2) crosses the x-axis by substituting y = 0 into the equation:

This is the x-intercept and can be plotted on the axes:

Complete the line by joining the y- and x-intercepts with a straight line:

Step 3
Find the point where the two lines cross.

Step 4
The x-coordinate of the point found in Step 3 gives the value of x that solves the simultaneous equations.

Step 5
The y-coordinate of the point found in Step 3 gives the value of y that solves the simultaneous equations.

The solution to the simultaneous equations is:

The slider below shows an example of solving simultaneous equations using the substitution method:
show

##### Note
WHAT IS THE EQUATION OF A STRAIGHT LINE?

The equation of a straight line is given by:

This is recognisable as the form of the simultaneous equations - this section deals with solving systems of linear equations.

Another way of writing linear equations such as this is in the standard form for the equation of a straight line:

where m is the slope of the line, and c is where the line crosses the y-axis.

WHAT IS THE SLOPE OF A LINE?

The slope, m, is given by the "rise over the run". More formally, it is given by:

Note: the Δ symbol (the Greek letter 'Delta') is often used to denote "change in".

For example, consider the linear equation below:

A slope of 3 means that the line goes up by 3 for every 1 it goes across:

Note A positive slope means the line slopes upwards from left to right.

Consider another example:

A slope of -2 means that the line goes down by 2 for every 1 it goes across:

Note A negative slope means the line slopes downwards from left to right.

WHAT IS THE Y-INTERCEPT?

The y-intercept, c, is the point where the line crosses the vertical y-axis.

For example, consider the linear equation below:

When plotted, the graph looks like:

With a small change to the linear equation, changing the y-intercept from 10 to 5, the line shifts downwards, crossing the y-intercept at 5. (Note: it is parallel to the original line. All lines with the same slope are parallel.)

Lines can also have negative y-intercepts, for example:

##### Top Tip
A FAST WAY TO DRAW A LINE: FIND THE INTERCEPTS

A quick method for plotting a linear function as a line is to find the y- and x-intercepts and to join them with a straight line.

To find the y-intercept, insert x = 0 into the linear equation.

To find the x-intercept, insert y = 0 into the linear equation.