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 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
x2+2x=−3x+6
Take each side of this equation and plot their graphs on the same coordinate grid:
y=x2+2x
and
y=−3x+6
These intersect in two places: (−6,24) and (1,3). Hence the solutions to the original equation are x=−6 and 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.