Follow Us on Twitter

How to Solve Simultaneous Equations Using Graphical Methods (Mathematics Lesson)

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:
Interactive Test
  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.