2.14.4 Using a graph
At IGCSE, Sometimes you need to use graphs to solve equations
- Solutions of a function are where the curve of a function crosses the x-axis
- If you are asked to do something like “Use the graph of \(y=fx\) to solve the given equation by
drawing a suitable line”- Rearrange the equation in the form \(fx=mx+c\) and then draw the straight line for \(y=mx+c\)
- Solutions are the x-coordinates where the line \(y=mx+c\) intersects the curve \(y=fx\)
Worked example:
The graph of \(y=x^2+5x\) for \(-2.5≤x≤2.5\) has been drawn for you .
By drawing suitable straight lines on the grid, solve the following equations .
\(x^2+\frac{1}{x}=-2\)
As \( {y=x}^2+\frac{1}{x}\) and \(x^2+\frac{1}{x}=-2\) So, \(y=-2\)
Draw the line \(y=-2\) and find the x-coordinate of the point of intersection of the line \(y=-2\) and the curve \(y=x^2+\frac{1}{x}\)
The x-coordinate of the point of intersection is -0.46 .
\(x^2+\frac{1}{x}+x-1=0\)
Rearrange the equation to get the equation of the curve on one side
\(x^2+\frac{1}{x}=-x+1\)
\(y=-x+1\)
Draw the line \(y=-x+1\) and find the x-coordinate of the point where \(y=-x+1\)
intersects the curve \(y=x^2+\frac{1}{x}\)
Here the gradient is -1 and y-intercept is 1
The x-coordinate of the point of intersection is -1.85 .
To solve simultaneous equations by using graphs
- Plot both equations on same set of axes
- Find where they intersect
- The x and y coordinates of the point of intersection are the x and y solutions of the simultaneous equations.
Worked example:
Solve the simultaneous equations
\(3x-2y=5\)
\(5x+y=9\)
Rearrange both equations in the form \(y=mx+c\)
\(y=\frac{3}{2}x+\frac{5}{2}\)
\(y=-5x+9\)
Plot both equations on the same set of axes
Find the point of intersection
The coordinates of the point of intersection are the solution of the simultaneous equations.
The point of intersection is 1, 4 .
So, the solution to the equations is \(x=1\) and \(y=4\) .
Worked example:
Solve the simultaneous equations
\(y=x^2+5x\)
\(y=4x-1\)
Plot both equations on the same set of axes
Find the point of intersection
The coordinate of the point of intersection is the solution of the simultaneous equations.
The approximate solutions from the graph are \((0.24, 1.24)\) and \((-4.24, -3.24)\) .