Hey, everyone. So throughout our discussion on lines, we've seen problems where they ask us to calculate the slope of a line by looking at the rise over the run between two points. But some problems, like the one we're going to work out down below, won't ask just for that. Some problems will ask us to look at the graph of a line and write its equation in a very specific way called the slope-intercept form. So that's what I want to talk about in this video. And what I'm going to show you is it's basically just a very specific way that we write an equation of two variables using two things we've already seen before independently. So we can use its slope and its intercept, specifically the y-intercept. Alright? So I'm going to show you this equation, and it's probably something that you've seen before. So let's get started.
The slope-intercept form of an equation is actually just an equation. That's y=mx+b. You've probably heard that at some point in a math class. This is just the slope-intercept form of a line. It's one of the most easy and straightforward ways to describe a line equation. So let's get started and talk about these two variables here. The m is basically just the slope, and we've already seen that before. We calculate that by using Δy over Δx, rise over run, or you can just use the sort of longer format here. So, in this equation, what we can see is that the rise over the run between these two points is 2/1, and so the slope is just 2. What about the y-intercept? Well, the b term over here is the y-intercept, and we've already talked about that separately. The y-intercept is basically just the y-value wherever the graph crosses the y-axis, and it's where x is equal to 0. So for instance, in this case, the graph crosses the y-axis right over here. The y-value at this point is 3. So that is the b term. Right? So that's the y-intercept. So to put these things together, y=mx+b, all you have to do is put the equation here has a slope of 2, so that goes in front of the x, and then the intercept is 3. So the equation of this line is just y=2x+3. That's how to describe this equation in slope-intercept form. That's really all there is to it. Alright?
So that's all there is to the slope-intercept form. Let's go ahead and take a look at another example. Alright? So, in this graph below, we're going to identify the slope, the slope, and the y-intercept, and then we're going to write the equation in slope-intercept form. So remember, slope-intercept form here is just going to be y=mx+b. In order to figure out m and b, we're going to have to take a look at the graph. Alright? So the easiest sort of value to start with is going to be the b term because you really just have to look at the graph and figure out where it crosses the y-axis. You don't have to calculate anything. So let's take a look at this graph. Where does it cross the y-axis? Well, it crosses right over here. So what's the y-value where this graph crosses the y-axis? It's just negative 3. So that's the b. It's just negative 3. That's all there is to it. You don't have to write the ordered pair. It's just the y-value. Alright?
So how do we calculate m? Well, for m, we're going to have to use rise over run. So we're just going to calculate this by using Δy over Δx. Or if we're given two points, you could just plug those points in. But, basically, if I have this point over here, what's the rise over the run to get to the next point? Well, if I take a look at this graph here, what happens is I've got all these different points that are all at the intersections. These little lines here, all these are valid points that I can pick. So what's the rise over the run? Well, to get from this point to the next one, all I have to do is go up 1 and then over 1. So up 1 over 1. So that's just a rise over the run of just 1. So the slope is 1. I just go 1 over and 1 up. So because the slope is just equal to 1, I basically just plug both of these things into this equation, and my equation is just equal to y=1x+-3. Just to simplify this, a lot of times that you'll see here is if there's just a one in front, you don't even write it. So it's just y=x-3. So this is the equation that describes this line in slope-intercept form, y=x-3. Alright? So, anyway, folks, thanks for watching. Let me know if you have any questions.