Lines through the origin

The idea

The straight line representing an equation of the form y = ax always passes through the origin (0, 0).

The line y = 3x

For the line y = 3x, the y of every point is 3 times its x. The points (−1, −3), (1, 3), (3, 9) and (4, 12) all lie on it. Now think about x = 0.

What do you think happens at x = 0 on the line y = 3x?

• The line passes through the origin (0, 0)
• The line passes through (0, 3)
• The line never meets the y-axis

Turn the line

Change a and watch the line y = ax. Where does it cross the y-axis?

Always through O

The straight line y = ax always passes through the origin (0, 0).

When x = 0, y = a × 0 = 0, whatever a is. So (0, 0) is on every line y = ax. Here are y = 3x and y = −2x: both go through O.

Notes

Every line of the form y = ax passes through the origin (0, 0).

Check yourself

Which line passes through the origin?

The line y = ax passes through the point (2, 10). What is a?

Answer: 5

10 = a × 2, so a = 5. The line is y = 5x.

For the line y = −4x, what is y when x = 0?

Answer: 0

−4 × 0 = 0. The line passes through the origin.

Which statement is true for every line y = ax?

• y = 5x + 1. When x = 0, y = 1. It crosses the y-axis at (0, 1).
• y = 5x — correct. Yes. When x = 0, y = 5 × 0 = 0.
• y = x − 2. When x = 0, y = −2. It crosses the y-axis at (0, −2).
• It passes through (0, 1). Only y = ax + 1 does that.
• It is a horizontal line. Only the line y = 0 is horizontal. Most lines y = ax are tilted.
• It passes through (0, 0) — correct. Yes. At x = 0, y = 0.