Solving equations using graphs

In a nutshell

One way to solve an equation is to use relevant graphs and see where they intersect. Any points of intersection will give a solution to the equation.

Solve equations using graphs

Solve an equation be making each side of the equation equal to $y$ , giving two equations of lines that can be plotted or sketched. The point where the two lines intersect is the solution.

Example 1

Solve the equation

$5x-17=2x-2$ 

To solve this using graphs, plot the graphs given by each side of the equation:

$y=5x-17$ 

and

$y=2x-2$

The point of intersection gives the solution. In particular, since you are looking for an $x$ value, the $x$ -coordinate of the point of intersection will be the solution.

Maths; Graphs; KS4 Year 10; Solving equations using graphs

These lines intersect at $(5,8)$ hence the solution to this equation is $\underline{x=5}$ .

An alternative method is to rearrange this equation such that one side is zero:

$3x-15=0$

Then plot the graph $y=3x-15$ and see what the $x$ -coordinate is when $y=0$ (since this is the other side of the rearranged equation above).

This method is applicable to non-linear graphs too.

Graphs in general

You can take any equation in one variable (for example in $x$ ) and use the graphs of each side to solve. Remember that it's the point(s) of intersection that give the solution(s).

Example 2

Using graphs, solve

$x^2+2x=-3x+6$

Take each side of this equation and plot their graphs on the same coordinate grid:

$y=x^2+2x$

and

$y=-3x+6$ 

Maths; Graphs; KS4 Year 10; Solving equations using graphs

These intersect in two places: $(-6,24)$ and $(1,3)$ . Hence the solutions to the original equation are $\underline{x=-6}$ and $\underline{x=1}$ .

Knowing this method helps you to visualise how many solutions there should be for a given equation.

Example 3

How many solutions are there to the following equation?

$\sin(x)=2$ 

Maths; Graphs; KS4 Year 10; Solving equations using graphs

The sine graph does not go up as high as $y=2$ , so there are zero solutions to this equation since there are zero points of intersection between $y=\sin(x)$ and $y=2$ .