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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 should be the y-intercept, the place where the line crosses the y-axis.

Well first of all, horizontal lines only have a y-intercept. They don’t intersect the x-axis cuz 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.

If we have the equation of a line, how do we find the intercepts? Here’s an equation, for example. Suppose we’re given in this equation, and we need to find the x and or the y intercept. Recall from the lessons on vertical and horizontal lines that the equation for the x axis is y equals 0.

And the equation for the, for the y axis is, is x equals 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 equals 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 equals 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 equals 0.

Then solve. We get 2x equals 3, divide by 2, and we get the positive fraction x equals three-halves, that’s the x-intercept of the line. Now we’ll plug in x equals zero to find the y intercept, simplify the math, divide by negative six, 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 equals 5 and y equals, y-intercept equals negative 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 comma 0 and 0 comma negative 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, if it has to be the x-intercept. Similarly, any point that has an x-coordinate of 0 has to be on the y-axis, so it has to be a y-intercept. 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 equals s, and the y-intercept equals 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 along 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 negative 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 the y-axis. Again, the rise and the run have equal absolute magnitude. So that ratio is a ratio of one, but it’s negative, so it has to be negative 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 triangle is if we have a slope of positive 1 or negative 1.

This obviously has a negative slope. So the slope has to be negative 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 equal 0 into the equation.

And we find the y-intercept by plugging in x equals 0. And the intercepts can be specified as points. So for example p comma 0 would be an x-intercept, and 0 comma q would be a y-intercept.

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