Solving Equations Graphically (Edexcel IGCSE Further Pure Maths)

Solving Equations Graphically

How can I solve equations graphically?

• A graph can be used to to help solve an equation like

  • Draw the graphs of and 

  • The solutions are the x-coordinates of the points of intersection

• This can be used when an equation is difficult or impossible to solve algebraically

  • The solutions found will usually be approximations rather than exact answers

  • The more accurate the graph, the more accurate the approximation

How can I estimate a solution by drawing a line on a graph?

• An exam question my ask you to estimate a solution by drawing a 'suitable' (or 'appropriate') straight line on a graph

• Often this will be a horizontal line

  • For example solving  for some constant
    

    • On a graph of , draw the line 

    • The solutions are the x-coordinates of any points of intersection

  • Finding roots by seeing where a graph crosses the x-axis is a special case of this

    • The x-axis is the horizontal line with equation  

• Sometimes it will be the line  

  • Draw this on the graph of  to find the solution(s) of

• But sometimes determining the line to draw will be more challenging

  • For example, 'By drawing an appropriate straight line on the graph of , estimate the root of the equation  '

  • We need to rewrite the equation in the form  , where  is the equation of a straight line

  • Take the exponential of both sides ('exp cancels log')

    • 

  • Take the cube root of both sides

    • 

  • Multiply both sides by 2

    • 

  • Add 3 to both sides

    • 

  • That equation is equivalent to  

    • it will have the same solutions

  • So we need to draw the line    on the graph of 

    • the x-coordinates of the points of intersection will give the solution(s) for 

    • But those are the same as the solution(s) for