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.
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.
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.
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.
About the Author
Based in San Diego, John Brennan has been writing about science and the environment since 2006. His articles have appeared in "Plenty," "San Diego Reader," "Santa Barbara Independent" and "East Bay Monthly." Brennan holds a Bachelor of Science in biology from the University of California, San Diego.