# 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.*

*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*

*$y=-3x+6$*

*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?*

*$\sin(x)=2$*

*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 $y=2$.*