Solving Equations Graphically (Edexcel IGCSE Further Pure Maths): Revision Note

Solving Equations Graphically

How can I solve equations graphically?

• A graph can be used to to help solve an equation like $\mathrm{f}\left(x\right)=\mathrm{g}\left(x\right)$

  • Draw the graphs of $y=\mathrm{f}\left(x\right)$ and $y=\mathrm{g}\left(x\right)$

  • 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 $\mathrm{f}\left(x\right)=k$ for some constant $k$
    

    • On a graph of $y=\mathrm{f}\left(x\right)$, draw the line $y=k$

    • 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  $\mathit{y}\mathbf{=}\mathbf{0}$

• Sometimes it will be the line  $\mathit{y}\mathbf{=}\mathit{x}$

  • Draw this on the graph of $y=\mathrm{f}\left(x\right)$ to find the solution(s) of $\mathrm{f}\left(x\right)=x$

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

  • For example, 'By drawing an appropriate straight line on the graph of $y=3+2{\mathrm{e}}^{-2x}$, estimate the root of the equation  $\ln {\left(x-3\right)}^3=-6x$'

  • We need to rewrite the equation in the form  $\mathrm{g}\left(x\right)=3+2{\mathrm{e}}^{-2x}$, where $y=\mathrm{g}\left(x\right)$ is the equation of a straight line

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

    • ${\left(x-3\right)}^3={\mathrm{e}}^{-6x}$

  • Take the cube root of both sides

    • $x-3={\mathrm{e}}^{-2x}$

  • Multiply both sides by 2

    • $2x-6=2{\mathrm{e}}^{-2x}$

  • Add 3 to both sides

    • $2x-3=3+2{\mathrm{e}}^{-2x}$

  • That equation is equivalent to  $\ln {\left(x-3\right)}^3=-6x$

    • it will have the same solutions

  • So we need to draw the line  $y=2x-3$  on the graph of  $y=3+2{\mathrm{e}}^{-2x}$

    • the x-coordinates of the points of intersection will give the solution(s) for  $2x-3=3+2{\mathrm{e}}^{-2x}$

    • But those are the same as the solution(s) for  $\ln {\left(x-3\right)}^3=-6x$