Solving Equations using Graphs (AQA GCSE Further Maths): Revision Note

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Updated on

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Solving Equations using Graphs

How do we use graphs to solve equations?

  • Solutions are always read off the x-axis

  • Solutions of f(x) = 0 are where the graph of y = f(x) crosses the x-axis

  • If asked to use the graph of y = f(x) to solve a different equation (the question will say something like “by drawing a suitable straight line”) then:

    • Rearrange the equation to be solved into f(x) = mx + c and draw the line y = mx + c

    • Solutions are the x-coordinates of where the line (y = mx + c) crosses the curve (y = f(x))

    • E.g. if given the curve for y = x3 + 2x2 + 1 and asked to solve x3 + 2x2  − x − 1 = 0, then;

      1. rearrange x3 + 2x2  − x − 1 = 0 to x3 + 2x2 + 1 = x + 2

      2. draw the line y = x + 2 on the curve y = x3 + 2x2 + 1

      3. read the x-values of where the line and the curve cross (in this case there would be 3 solutions, approximately x = -2.2, x = -0.6 and x = 0.8);

2-15-2-solving-equations-using-graphs
  • Note that solutions may also be called roots

Examiner Tips and Tricks

  • If solving an equation, give the x values only as your final answer

  • If solving a pair of linear simultaneous equations give an x and a y value as your final answer

  • If solving a pair of simultaneous equations where one is linear and one is quadratic, give two pairs of x and y values as your final answer

Worked Example

The graph of y equals x cubed plus x squared minus 3 x minus 1 is shown below.
Use the graph to estimate the solutions of the equation x cubed plus x squared minus 4 x equals 0. Give your answers to 1 decimal place.

Cubic-Linear-Intersections-(before), IGCSE & GCSE Maths revision notes

We are given a different equation to the one plotted so we must rearrange it to f open parentheses x close parentheses equals m x plus c (where f open parentheses x close parentheses is the plotted graph)

table attributes columnalign right center left columnspacing 0px end attributes row cell x cubed plus x squared minus 4 x end cell equals 0 end table

table row blank blank cell plus x minus 1 space space space space space space space space space space space space space space space space space space space space space space space space plus x minus 1 space space end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell x cubed plus x squared minus 3 x plus 1 end cell equals cell x minus 1 end cell end table

Now plot y equals x minus 1 on the graph- this is the solid red line on the graph below

Cubic-Linear-Intersections-(after), IGCSE & GCSE Maths revision notes

The solutions are the x coordinates of where the curve and the straight line cross so

bold italic x bold equals bold minus bold 2 bold. bold 6 bold comma bold space bold space bold italic x bold equals bold 0 bold comma bold space bold space bold italic x bold equals bold 1 bold. bold 6

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Jamie Wood

Author: Jamie Wood

Expertise: Maths Content Creator

Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

Dan Finlay

Reviewer: Dan Finlay

Expertise: Maths Subject Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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