Understanding the X-Intercept and Y-Intercept of a Linear Equation

Finding the x- and y-intercepts of an equation are important skills you’ll need in math and the sciences. For some problems, this may be more complicated; fortunately, for linear equations it just couldn’t be simpler. A linear equation will only ever have, at most, one x-intercept and one y-intercept.

X-Intercept

A linear equation has the form y = mx + b, where M and B are constants. The x-intercept is the point where the line crosses the x-axis. By definition, the y-value of a linear equation when it crosses the x-axis will always be 0, since the x-axis is stationed at y = 0 on a graph. Consequently, to find a y-intercept, just substitute 0 for y and solve for x. This will give you the value of x at the x-intercept.

Y-Intercept

The y-intercept is the point at which the line crosses the y-axis; the value of x must be 0 at the y-intercept, because the y-axis is stationed at x = 0 on the graph. Consequently, to find the y-intercept, substitute 0 for x in your equation and calculate y. For equations of the form y = mx + b, this is especially easy; if x = 0, the first term (m times x) will be 0, so y will equal b. Thus, the constant b in a linear equation is the value of y at the y-intercept, while the constant m is the slope of the line — the larger m is, the steeper the slope.

Equations without Intercepts

Some equations do not have x- or y-intercepts; this usually happens when x or y are constant. For example, the equation y = 5 does not and cannot have an x-intercept, since y will never be equal to 0. Similarly, the equation x = 5 does not have a y-intercept as x will never be equal to 0. Both of these types of equations are flat lines that have no slope; the first one is perfectly horizontal, while the other is perfectly vertical.

Example

Here’s an example to illustrate how you can find x- and y-intercepts.

Example: Fine the x- and y-intercepts of the equation y = 10x – 12

To find the x-intercept, substitute y = 0 then solve.

0 = 10x – 12 12 = 10x x = 12 / 10 = 6/5. (or 1.2)

Therefore, the x-intercept is 6/5. Since this equation is in the form y = mx + b, and b is the value of y at the y-intercept, you also know the y-intercept must be -12.