Solving Equations Graphically

How can I solve equations graphically?

A graph can be used to to help solve an equation like $f (x) = g (x)$
- Draw the graphs of $y = f (x)$ and $y = g (x)$
- 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 $y = f (x)$ , 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 $y = 0$
Sometimes it will be the line
- Draw this on the graph of $y = f (x)$ to find the solution(s) of $f (x) = x$
But sometimes determining the line to draw will be more challenging
- For example, 'By drawing an appropriate straight line on the graph of $y = 3 + 2 e^{- 2 x}$ , estimate the root of the equation $\ln {(x - 3)}^{3} = - 6 x$ '
- We need to rewrite the equation in the form $g (x) = 3 + 2 e^{- 2 x}$ , where $y = g (x)$ is the equation of a straight line
- Take the exponential of both sides ('exp cancels log')
  - ${(x - 3)}^{3} = e^{- 6 x}$
- Take the cube root of both sides
  - $x - 3 = e^{- 2 x}$
- Multiply both sides by 2
  - $2 x - 6 = 2 e^{- 2 x}$
- Add 3 to both sides
  - $2 x - 3 = 3 + 2 e^{- 2 x}$
- 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 $2 x - 3 = 3 + 2 e^{- 2 x}$
  - But those are the same as the solution(s) for $\ln {(x - 3)}^{3} = - 6 x$

Exam Tip

Be extra careful when drawing graphs on 'estimate solutions by using a graph' questions
- The accuracy of your answer will depend on the accuracy of your drawing
- Use a ruler for straight lines

Worked example

A graph of $y = 2^{(\frac{x}{3} + 1)} - 1$ in the interval $0 \leq x \leq 6$ is shown in the following diagram

Graph of exponential function

By drawing a suitable straight line on the grid, show that the equation $\log_{2} {(3 x - 1)}^{2} - \frac{2}{3} x = 2$ has a root in the interval $0 \leq x \leq 6$ , and obtain an estimate for the value of that root.

Be careful here – we cannot just draw the horizontal line $y = 2$
That would only work if we had the graph of $y = \log_{2} {(3 x - 1)}^{2} - \frac{2}{3} x$

Instead we must work on rearranging the equation
Start by getting the logarithm alone on the left-hand side

$\log_{2} {(3 x - 1)}^{2} = \frac{2}{3} x + 2$

Use laws of logarithms to bring the power down in front of the logarithm
Then divide both sides of the equation by 2

$\begin{array}{rcl} 2 \log_{2} (3 x - 1) & = & \frac{2}{3} x + 2 \\ \log_{2} (3 x - 1) & = & \frac{x}{3} + 1 \end{array}$

Now take both sides to the power of 2
This will cancel the logarithm on the left-hand side ('exp cancels log')

$3 x - 1 = 2^{(\frac{x}{3} + 1)}$

Finally subtract 1 from both sides

$3 x - 2 = 2^{(\frac{x}{3} + 1)} - 1$

Now the right-hand side is the function that is graphed on the diagram
So we need to draw the straight line $y = 3 x - 2$
We can estimate the root by considering the x-coordinate of the point of intersection

Solution graph for question

root: $x \approx 1.2$

Solving Equations Graphically (Edexcel IGCSE Further Maths)

Revision Note