Solving simultaneous equations using graphs

​​In a nutshell

Simultaneous equations are a set of equations that, when worked with together, can be solved. One way to solve simultaneous equations is to use graphs. Equations of lines and graphs of lines represent the same information; the graphs offer a pictorial representation of the equations.

"Simultaneous"

The word "simultaneous" means that things happen at the same time. Hence, when you solve simultaneous equations, you solve equations together.

Example 1

Two simultaneous equations are given below:

$$2x+y=7$$

$$4x-2y=18$$

You don't need to solve these but you can check that $$\underline{x=4}$$ and $$\underline{y=-1}$$. This is the solution to these simultaneous equations. 

Note: To check this solution, insert the values for $$x$$ and $$y$$ into the equations to see if they give the numbers on the right-hand side.

Using graphs

You can plot straight line graphs when they are in the form $$y=mx+c$$. Any point of intersection between the lines gives the solution to the simultaneous equations.

Example 2

Using graphs, solve the simultaneous equations

$$2x+y=7$$​​

$$4x-2y=18$$​​

Rearranging these equations gives:

$$y=-2x+7$$

$$y=2x-9$$

You can now plot these two lines:

​​

These two lines intersect at one point. The solution to these simultaneous equations is given by the coordinates of this point. See that the $$x$$​-coordinate is $$\underline4$$​ and the $$y$$​-coordinate is $$\underline{-1}$$​. This agrees with what you saw in the example above.

Using non-linear graphs

This method of plotting two graphs and using the point(s) of intersection applies also to non-linear graphs. Once you know how to plot non-linear graphs, you can apply this method to solve non-linear simultaneous equations.

Note: Linear simultaneous equations only have, at most, one solution because lines can only intersect in one place. Non-linear simultaneous can have more solutions because their graphs may intersect multiple times.