Graphical solutions
Objective. Interpret the intersection of two lines as a shared solution.
- 1. Learn
- 2. Worked example
- 3. Practice
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Learn: the key idea
A graphical solution to simultaneous equations finds the point where two straight lines cross, since that point's coordinates satisfy both equations at once. To solve graphically, plot both lines accurately on the same axes, usually using a table of values or the gradient-intercept form, then read off the coordinates of the intersection point. For example, the lines y = x + 1 and y = −x + 5 cross where x + 1 = −x + 5, giving 2x = 4, so x = 2 and y = 3, matching the point (2, 3) seen on the graph. If the lines are parallel, they never intersect, meaning the simultaneous equations have no solution.
Worked example
The lines y = 2x − 1 and y = −x + 5 are drawn on a graph. Find the coordinates of their intersection by solving the simultaneous equations represented.
- 1Set the expressions equal since both equal y at the intersection: 2x − 1 = −x + 5.
- 2Add x to both sides: 3x − 1 = 5.
- 3Add 1 to both sides and divide by 3: 3x = 6, so x = 2.
- 4Substitute x = 2 into y = 2x − 1: y = 4 − 1 = 3.
Answer: (2, 3)
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Practice
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Start practisingCommon mistake. Pupils sometimes read the intersection point's coordinates in the wrong order, swapping the x and y values from the graph. Always state coordinates as (x, y), checking the reading against both original equations.
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