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Graphical Solutions

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

How do we use graphs to solve linear simultaneous equations?

  • Plot both equations on the same set of axes using straight line graphs y = mx + c
  • Find where the lines intersect (cross)
    • The x and y solutions to the simultaneous equations are the x and y coordinates of the point of intersection
  • e.g. to solve 2x - y = 3 and 3x + y = 4 simultaneously, first plot them both (see graph)
    • find the point of intersection, (2, 1)
    • the solution is x = 2 and y = 1

Solving Equations Graphically Notes Diagram 1, A Level & AS Level Pure Maths Revision Notes

How do we use graphs to solve simultaneous equations where one is quadratic?

  • e.g. to solve y = x2 + 4x − 12 and y = 1 simultaneously, first plot them both (see graph)
    • find the two points of intersection (by reading off your scale), (-6.1 , 1) and (-2.1, 1) to 1 decimal place
    • the solutions from the graph are approximately x = -6.1 and y = 1 and x = 2.1 and y = 1
      • note their are two pairs of x, y solutions 
      • to find exact solutions, use algebra

Solving Equations Graphically Notes Diagram 2, A Level & AS Level Pure Maths Revision Notes

Exam Tip

  • 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 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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