Coordinate Geometry: X and Y Intercepts

Our study of coordinate geometry continues! Now we turn to x and y intercepts, so check out the video below — and remember that the transcript is right beneath it!

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Transcript: X and Y Intercepts

Now we can talk about intercepts. The intercepts of a line are the points at which the line crosses the x and y-axes. These are known specifically as the x-intercept and the y-intercept. So for example, this point right here would be the x-intercept, the place where the line crosses the x-axis. And this would be the y-intercept, the place where the line crosses the y-axis.

x and y intercepts - magoosh

Image by Yu_Peri

Well, first of all, horizontal lines have only a y-intercept. They don’t intersect the x-axis because they’re parallel to it. Similarly, vertical lines only have an x-intercept. They don’t intersect the y-axis because they’re parallel to it. Any line that passes through the origin has both an x-intercept and y-intercept of zero.

So that’s the only time that a slanted line would have its x-intercept and y-intercept at exactly the same point. Usually what happens is that if a slanted line doesn’t pass through the origin — it has an x-intercept in one place, and a y-intercept in another place. It has two different intercepts, and these points are enough to determine a unique line.

Equation of a Line: How to Find X and Y Intercepts

If we have the equation of a line, how do we find the intercepts?

Here’s an equation for example, suppose we’re given this equation and we need to find the x and/or the y-intercept. Recall from the lesson on Vertical and Horizontal Lines that the equation for the x-axis is y=0, and the equation for he y-axis is x=0. So in other words, any point on the x-axis has a y coordinate of 0.

And any point on the y-axis has an x coordinate of 0. That’s a very deep idea. What this means is if we plug y = 0 into the equation, we’re gonna get a point on the x-axis, we’re gonna get the x-intercept. Similarly, if we plug x = 0 into the equation, we’re automatically gonna get a point on the y-axis. In other words, we’re gonna get the y-intercept.

So we’ll just do this. First we’ll solve for the x-intercept. We’ll plug in y = 0. Then solve, we get 2x = 3 divide by 2, and we get the positive fraction, x = 3 halves. That’s the x-intercept of the line.

Now we’ll plug in x = 0 to find the y intercept. Simplify the math, divide by -6, we get the fraction, negative one-half, that is the y-intercept of the line. The intercepts can be stated as equations So we could say x-intercept = 5, and y-intercept = -3 for some point. We could also state those in a very different way, we could state them as points.

So we could say line A passes through (5,0) and (0, -3). And notice we’re giving intercepts there. Because any point on the line that has a y-intercept of 0 has to be on the x-axis, it has to be the x-intercept. Similarly any point that has an X x-coordinate of 0 has to be on the y-axis, so it has to be a y-intercept.

Practice Problem

Here’s a practice problem, pause the video and then we’ll talk about this.

Okay, so we’re given the x and the y-intercepts of this line. And notice that they have the same numerical values. So the x-intercept = s, and the y-intercept = s, and we want to know what’s the slope of the line. Well let’s think about this.

First of all, let’s pretend that s is a positive number, we’ll look at that case first. If s is a positive number, then we go a positive direction along the x-axis and a positive direction along the y-axis. And so the line passes through the first quadrant like that. And notice that the rise and the run have equal magnitude.

So if you take the absolute value of the rise and the absolute value of the run, they’re identical, and of course the slope is negative, so essentially we get a slope of -1. So that’s what happens if s is positive. What happens if s is negative? Well, if s is negative, we get something very similar.

We get a triangle in the third quadrant, so now we’ve gone in a negative direction along the x-axis and y-axis. Again, the rise and the run have equal absolute magnitude. So that ratio is a ratio one, but it’s negative. So, it has to be -1. Notice both of these triangles incidentally are 45, 45, 90 triangles.

And so of course, the only time we get a 45, 45, 90 triangles if we have a slope of positive one and a -1. This obviously has a negative slope. So the slope has to be -1. In summary, the x-intercept of a line is the point where the line intercepts the x-axis, similarly for the y-intercept.

We find the x-intercept from an equation by plugging y = 0 into the equation, and we find the y-intercept by plugging in x = 0. And the intercepts can be specified as points, so for example (p, 0) would be and x-intercept and (0, q) would be an y-intercept.

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