The intercepts of a linear equation are the places where the axes catch the pass thrown by the linear equation. This is our effort to make linear equations seem remotely athletic. In reality, they have about as much physical ability as Tim Tebow. Oh, snap.
(Shmoop is strongly partisan about football, in case you couldn't tell.)
In non-sports-analogy terms, the intercepts are the spots at which the axes and the graph of the linear equation overlap one another. The x-intercept is the place where the graph hits the x-axis, and the y-intercept is the place where the graph hits the y-axis. It would be awfully confusing if it were the other way around.
A linear equation may have one or two intercepts. Sometimes either the x-intercept or the y-intercept doesn't exist, or so intercept atheists would have you believe.
Knowing both intercepts for a linear equation is enough information to draw the graph, provided the intercepts aren't 0. If they are 0, then our graph could be drawn any which way.
Sample Problems
Draw the graph of the linear equation with x-intercept 3 and y-intercept 4.
First we draw points at the intercepts:
Then we connect the dots:
If the graph goes through the origin (0, 0), then both of the intercepts are 0 and we don't have enough information to draw the graph. We even tried calling 411, but they acted as if they had no idea what we were talking about.