Solving Equations from Graphs (Edexcel IGCSE Maths B): Revision Note

Solving equations using graphs

How do I find the coordinates of points of intersection?

• Plot two graphs on the same set of axes

  • The points of intersection are where the two lines meet

• For example, plot y  = x² + 3x + 1 and y = 2x + 1 on the same axes

  • They meet twice, as shown

  • The coordinates of intersection are (-1, -1) and (0, 1)

How do I solve simultaneous equations graphically?

• The x and y solutions to simultaneous equations are the x and y coordinates of the point of intersection

• For example, to solve 2x - y = 3 and 3x + y = 7 simultaneously

  • Rearrange them into the form y = mx + c

    • y = 2x - 3 and y = -3x + 7

  • Use a table of values to plot each line

  • Find the point of intersection, (2, 1)

  • The solutions are therefore x = 2 and y = 1

How do I use graphs to solve equations?

• This is easiest explained through an example

• You can use the graph of $y=x^2-4x-2$ to solve the following equations

  • $x^2-4x-2=0$

    • The solutions are the two x-intercepts

    • This is where the curve cuts the x-axis (also called roots)

  • $x^2-4x-2=5$

    • The solutions are the two x-coordinates where the curve intersects the horizontal line $y=5$ 

  • $x^2-4x-2=x+1$

    • The solutions are the two x-coordinates where the curve intersects the straight line $y=x+1$

    • The straight line must be plotted on the same axes first

• To solve a different equation like $x^2-4x+3=1$, if you are already given the graph of an equation, e.g. $y=x^2-4x-2$

  • add / subtract terms to both sides to get "given graph = ..."

    • For example, subtract 5 from both sides

      • $x^2-4x-2=-4$

      • You can now draw on the horizontal line $y=-4$ and find the x-coordinates of the points of intersection