284. The Constant Lines · x=0, y=0, x=a, y=a on the coordinate plane

Try to break it

Drag P horizontally to change a — the vertical line x = a slides left and right but always stays parallel to the y-axis. Drag Q vertically to change b — the horizontal line y = b slides up and down parallel to the x-axis. Set a = 0 and the line x = a merges with the y-axis; set b = 0 and the line y = b merges with the x-axis.

How you build it

Draw the constant lines x = a and y = b.

• Line tool: draw a vertical line (two clicks at the same x). It is x = a — parallel to the y-axis, every point has the same x.
• Line tool: draw a horizontal line (two clicks at the same y). It is y = b — parallel to the x-axis, every point has the same y.

The proof, step by step

Prove that x = a is a vertical line and y = b is a horizontal line.

• The equation x = a means every point on this line has an x-coordinate equal to 'a'.
• The y-axis is defined by x = 0. Since x = a has a constant x-value, it is a vertical line.
• All vertical lines are parallel to each other. Thus, x = a is parallel to the y-axis (x=0).
• Similarly, y = b means every point has y-coordinate 'b', making it a horizontal line parallel to the x-axis (y=0).

Worked example

In a Cartesian plane, the line x = -4 is parallel to which axis? A) x-axis B) y-axis C) Both D) Neither

The equation x = -4 represents a vertical line where every point has an x-coordinate of -4. Vertical lines are always parallel to the y-axis (which is x=0). Therefore, x = -4 is parallel to the y-axis.

• x-axis
• y-axis — correct
• Both axes
• Neither axis