Solving Equations from Graphs (Cambridge (CIE) IGCSE Maths)

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Written by: Mark Curtis

Reviewed by: Dan Finlay

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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)

Points of intersection between a curve and a line

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

Solving simultaneous equations graphically

How do I use graphs to solve equations?

This is easiest explained through an example
You can use the graph of $y = x^{2} - 4 x - 2$ to solve the following equations
- $x^{2} - 4 x - 2 = 0$
  - The solutions are the two x-intercepts
  - This is where the curve cuts the x-axis (also called roots)
- $x^{2} - 4 x - 2 = 5$
  - The solutions are the two x-coordinates where the curve intersects the horizontal line $y = 5$
- $x^{2} - 4 x - 2 = x + 1$
  - 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 $x^{2} - 4 x + 3 = 1$ , if you are already given the graph of an equation, e.g. $y = x^{2} - 4 x - 2$
- add / subtract terms to both sides to get "given graph = ..."
  - For example, subtract 5 from both sides
    - $x^{2} - 4 x - 2 = - 4$
    - You can now draw on the horizontal line $y = - 4$ and find the x-coordinates of the points of intersection

Examiner Tips and Tricks

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

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

This equals zero, so the x-intercepts 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

x = -1.12 and x = 1.12

A range of solutions are accepted, such as "between 1.1 and 1.2"
Solutions must be ± of each other (due to the symmetry of quadratics)

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

This 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

x = -0.5 and x = 0.5

The solutions here are exact

Worked Example

The graph of $y = x^{3} + x^{2} - 3 x - 1$ is shown below.

Use the graph to estimate the solutions of the equation $x^{3} + x^{2} - 4 x = 0$ .

Give your answers to 1 decimal place.

We are given a different equation to the one plotted so we must rearrange it to $graph = m x + c$ , in this case $x^{3} + x^{2} - 3 x - 1 = m x + c$

$\begin{array}{rcl} x^{3} + x^{2} - 4 x & = & 0 \end{array}$

$\begin{array}{rcl} + x - 1 + x - 1 \end{array}$

$\begin{array}{rcl} x^{3} + x^{2} - 3 x + 1 & = & x - 1 \end{array}$

Now plot $y = x - 1$ on the same axes

The graph of y = x^3 + x^2 - 3x - 1 and the line y = x - 1

The solutions are the $x$ -coordinates of where the curve and the straight line intersect

$x = - 2.6, x = 0, x = 1.6$

Last updated: 17 October 2024

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Solving Equations from Graphs (Cambridge (CIE) IGCSE Maths)

Revision Note

Solving Equations Using Graphs

How do I find the coordinates of points of intersection?

How do I solve simultaneous equations graphically?

How do I use graphs to solve equations?

Examiner Tips and Tricks

Worked Example

Worked Example

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Author: Mark Curtis

Author: Dan Finlay