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Graphical Solutions (Edexcel IGCSE Maths)
Revision Note
Solving Equations Using Graphs
How do we use graphs to solve equations?
- Solutions are always read off the x-axis
- Solutions of f(x) = 0 are where the graph of y = f(x) crosses the x-axis
- If asked to use the graph of y = f(x) to solve a different equation (the question will say something like “by drawing a suitable straight line”) then:
- Rearrange the equation to be solved into f(x) = mx + c and draw the line y = mx + c
- Solutions are the x-coordinates of where the line (y = mx + c) crosses the curve (y = f(x))
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E.g. if given the curve for y = x3 + 2x2 + 1 and asked to solve x3 + 2x2 − x − 1 = 0, then;
- rearrange x3 + 2x2 − x − 1 = 0 to x3 + 2x2 + 1 = x + 2
- draw the line y = x + 2 on the curve y = x3 + 2x2 + 1
- read the x-values of where the line and the curve cross (in this case there would be 3 solutions, approximately x = -2.2, x = -0.6 and x = 0.8);
- Note that solutions may also be called roots
How do we use graphs to solve linear simultaneous equations?
- Plot both equations on the same set of axes using straight line graphs y = mx + c
- Find where the lines intersect (cross)
- The x and y solutions to the simultaneous equations are the x and y coordinates of the point of intersection
- e.g. to solve 2x - y = 3 and 3x + y = 4 simultaneously, first plot them both (see graph)
- find the point of intersection, (2, 1)
- the solution is x = 2 and y = 1
How do we use graphs to solve simultaneous equations where one is quadratic?
- e.g. to solve y = x2 + 4x − 12 and y = 1 simultaneously, first plot them both (see graph)
- find the two points of intersection (by reading off your scale), (-6.1 , 1) and (-2.1, 1) to 1 decimal place
- the solutions from the graph are approximately x = -6.1 and y = 1 and x = 2.1 and y = 1
- note there are two pairs of x, y solutions
- to find exact solutions, use algebra
Examiner Tip
- If solving an equation, give the x values only as your final answer
- If solving a pair of linear simultaneous equations give an x and a y value as your final answer
- If solving a pair of simultaneous equations where one is linear and one is quadratic, give two pairs of x and y values as your final answer
Worked example
The graph of is shown below.
Use the graph to estimate the solutions of the equation . Give your answers to 1 decimal place.
We are given a different equation to the one plotted so we must rearrange it to (where is the plotted graph)
Now plot on the graph- this is the solid red line on the graph below
The solutions are the coordinates of where the curve and the straight line cross so
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