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Solving equations using graphs

Solving equations using graphs

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Explainer Video

Tutor: Toby

Summary

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 yyy, 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−25x-17=2x-25x−17=2x−2​​


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

y=5x−17y=5x-17y=5x−17​​

and

y=2x−2y=2x-2y=2x−2


The point of intersection gives the solution. In particular, since you are looking for an xxx value, the xxx-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)(5,8)(5,8) hence the solution to this equation is x=5‾\underline{x=5}x=5​.


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

​​3x−15=03x-15=03x−15=0


Then plot the graph y=3x−15y=3x-15y=3x−15 and see what the xxx-coordinate is when y=0y=0y=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 xxx) 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+6x^2+2x=-3x+6x2+2x=−3x+6


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

y=x2+2xy=x^2+2xy=x2+2x

and

y=−3x+6y=-3x+6y=−3x+6​​

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


These intersect in two places: (−6,24)(-6,24)(−6,24) and (1,3)(1,3)(1,3). Hence the solutions to the original equation are x=−6‾\underline{x=-6}x=−6​ and x=1‾\underline{x=1}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\sin(x)=2sin(x)=2​​

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


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

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Drawing a graph of a linear equation

Unit 1

Drawing a graph of a linear equation

Estimating values using linear graphs

Unit 2

Estimating values using linear graphs

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Solving equations using graphs

Unit 3

Solving equations using graphs

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FAQs - Frequently Asked Questions

How do points of intersection give solutions to equations?

The x-coordinate is the x value of the solution. Sometimes a y value will be part of a solution - this is given by the y-coordinate.

How can you use graphs to solve an equation?

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.

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