Drawing linear graphs

Maths · Coordinate geometry. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A linear equation such as y = 2x − 3 or 3x + 4y = 12 gives a straight line. Every point on the line has coordinates that fit the equation, and no other point does.

To draw the line, make a table of values: choose a few values of x, work out y for each, plot the points and join them with a ruler. Two points are enough for a straight line, but a third point is a useful check.

A quick method is to find the intercepts. Put x = 0 to see where the line crosses the y-axis, and put y = 0 to see where it crosses the x-axis.

Rules to remember

A point lies on a line if its coordinates make the equation true
y-intercept: put x = 0; x-intercept: put y = 0
y = mx + c crosses the y-axis at (0, c)
y = mx (no constant term) passes through the origin
x = a is a vertical line through (a, 0); y = b is a horizontal line through (0, b)
Two lines cross at the point whose coordinates fit both equations (the solution of the simultaneous equations)

Common mistake

Mixing up the intercepts: students put y = 0 and then say that is where the line crosses the y-axis. On the y-axis x is 0; on the x-axis y is 0.

Worked example

Where does the line 5x − 2y = 20 cross the axes?

Put x = 0: −2y = 20, so y = −10. It crosses the y-axis at (0, −10).
Put y = 0: 5x = 20, so x = 4. It crosses the x-axis at (4, 0).
Check with a third point, x = 2: 10 − 2y = 20, so y = −5, and (2, −5) is in line with the other two.

Answer: (4, 0) and (0, −10)

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