Solving Equations from Graphs (Cambridge International IGCSE Mathematics)

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 2*x - 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 $x^{2}-4x+3=1$

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

    • For example, subtract 5 from both sides

      • $x^{2}-4x-2=-4$
      • This can now be done as above

Exam Hint

• When solving equations in x, only give x-coordinates as final answers

  • Include the y-coordinates if solving simultaneous equations

Worked Example Use the graph of $y=10-8x^{2}$ shown to estimate the solutions of each equation given below.

(a) $10-8x^{2}=0$

Answer:

The function equals zero, so the x-intercepts of its graph are the solutions Read off the values where the curve cuts the x-axis Use a suitable level of accuracy (no more than 2 decimal places from the scale of this graph)

-1.12 and 1.12

These are the two solutions to the equation

Final answer: x = -1.12 and x = 1.12

(b) $10-8x^{2}=8$

Answer:

The function equals 8, so draw the horizontal line y = 8 *Find the x-coordinates where this cuts the graph *

-0.5 and 0.5

These are the two solutions to the original equation

Final answer: x = -0.5 and x = 0.5