Coordinate Axes Equations
Imagine a blank sheet of paper. You want to describe exactly where a point is. The most natural way is to draw two perpendicular lines — one horizontal, one vertical — and measure distances from them. Those two lines are the coordinate axes.
The horizontal line is the x-axis. The vertical line is the y-axis. Where they cross is the origin, the point (0,0).
Now, here's the key idea: every point on the x-axis has a y-coordinate of 0. Why? Because to be on that horizontal line, you haven't moved up or down at all — your vertical distance from the axis is zero. Similarly, every point on the y-axis has an x-coordinate of 0 — you haven't moved left or right from the vertical line.
That's the entire intuition. Let's make it precise.
The Equations
x-axis: y=0
y-axis: x=0
That's it. These are the equations of the coordinate axes.
- y=0 means: "all points where the y-coordinate is zero, regardless of x." That's the entire horizontal line through the origin.
- x=0 means: "all points where the x-coordinate is zero, regardless of y." That's the entire vertical line through the origin.
A common mistake is to think y=0 means "the point (0,0)". No — it means every point with y=0, like (5,0), (−3,0), (100,0), etc. It's a whole line, not a single point.
Why This Matters
These two equations are the foundation for everything in coordinate geometry. Every other line, curve, or shape is described relative to these axes. For example:
- The line y=2 is a horizontal line parallel to the x-axis, shifted up by 2 units.
- The line x=−3 is a vertical line parallel to the y-axis, shifted left by 3 units.
So when you see y=0 or x=0, think: "the original, unshifted axes themselves." …