The x-intercept is the point at which a line crosses the x-axis; i.e. the point at which y = 0. The y-intercept is the point at which a line crosses the y-axis; that is, the point at which x = 0. These concepts depend upon writing a linear equation using variables x and y, which is both standard and implicit in our identification of such an equation with the straight line that is its graph.
To find the x-intercept, set y = 0 and solve the equation. For example, to find the x-intercept of 5y - 2x = 10:
5(0) - 2x = 10
-2x = 10
x = - 5
Thus, the x-intercept, or the point at which the line crosses the horizontal axis, is (- 5, 0).
To find the y-intercept, set x = 0 and solve the equation. For example to find the y-intercept of 5y - 2x = 10:
5y - 2(0) = 10
5y = 10
y = 2
Thus, the y-intercept, or the point at which the line crosses the vertical axis, is (0, 2).
Hence, to find the intercept of either variable, set the other variable equal to 0 and solve for the original variable.
Graphing Using Intercepts
As observed in the last section, we only really need two points to graph a line. Usually, the two easiest points to find are the x-intercept and the y-intercept. Once these have been found, we can plot them, draw a straight line connecting them, and extend the line at either end. Here is a graph of the equation 5y - 2x = 10, drawn using intercepts:
Of course, it is useful to test a point on the line to make sure it satisfies the equation; since we are using only two points, there is more room for error.
It is important to point out that, no matter what technique we use to graph an equation, the graph of the equation is always the same -- all techniques will yield the exact same graph.