Hey, everyone. So, in previous videos, we saw how to write equations in point-slope form whenever we were given information about the slope and a point that the line passed through \(x_1, y_1\). What I'm going to show you is that in some problems like this one we're going to work out down here, sometimes you won't be given the slope, that \(m\) value, and instead, you'll be given information about 2 of the points. And what I'm going to show you is that we're still going to use the point-slope form for this, and it's actually going to turn out to be very similar to the problems we've already done before, but there's just one extra step. So let's go ahead and get started here. Alright?

So, I just want to remind you that we're not going to use the slope-intercept form \( y = mx + b \) for these types of problems because in those problems, we're usually given some information about \(m\) and \(b\), and we're asked for the graph, or you're asked for \(m\) and \(b\). We're not told anything about the y-intercept in this kind of problem. All we're told through is that it passes through these two points, \((-1, -5)\) and \((2, 4)\). So we don't use \( y = mx + b \), and instead, we're still going to use the point-slope form because we have some information about the points. So I'm going to write my equation \(y - y_1 = m (x - x_1)\). Remember, this equation, I need three numbers here. I need \(y_1\), \(x_1\), and I need the slope. Now in some problems, the slope was already given to us in previous problems, but in this case, it actually wasn't. So because we don't know what \(m\) is directly, we want to go find that first. So how do I go and calculate \(m\)? Well, remember that the \(m\) is really just the slope. It's the rise over the run. So, basically, you just use these two equations over here, \(\Delta y / \Delta x \) or \(y_2 - y_1 / x_2 - x_1\). So I'm going to use this \(y_2 - y_1 / x_2 - x_1\) because I'm told information about 2 of the points, and I don't have these things sort of graphed out already. Alright? So remember, I want to pick my points so that the \(x_1, y_1\) is the leftmost point. Again, it doesn't matter which point you pick. It'll still be the same no matter what. So this is going to be my \(x_1, y_1\), and this is going to be my \(x_2, y_2\). Alright? So here, what I've got here to calculate the slope is \(y_2 - y_1\). So, in this case, I've got \(4 - (-5)\). So I've got \(4 - (-5)\) over here, and then I've got divided by \(x_2 - x_1\). So this is going to be \(2 - (-1)\). \(2 - (-1)\). What I end up getting over here is I'm getting \(9/3\), which actually just gives me an \(m\) of \(3\). So that's my \(m\) term over here. I've got what my slope is, and I'm just going to pop that right back into this equation. But I'm not done yet because now I just need my \(x_1\) and \(y_1\). Now again, it doesn't matter which point you pick as your \(x_1, y_1\). So you can use these numbers and pop them into this equation over here for \(x_1, y_1\), or you could use these. It really just depends on which ones you picked. Now because I've already picked my \(x_1, y_1\) to be these numbers, I'm just going to go ahead and use these numbers. Okay?

So what I'm going to do here is I'm going to do \(y - (-5) = 3(x - (-1))\). And if I go ahead and sort of just draw a little box around this, this actually ends up being my final equation. So this is the equation of this line that passes through the points, and that's really all there is to it in point-slope form. Again, if you use these numbers, you would have gotten a slightly different form, and that's perfectly fine because it actually ends up being the same equation. So now, what we're going to do is we're just going to graph this. Alright? Well, graphing this is actually pretty straightforward because remember, we always need 2 points to graph a line, and we already know what they are. This is just \((-1, -5)\) and then \( (2, 4) \) over here. So all you have to do is just connect these lines or connect these points with a straight line like this. If you have a straight edge, that's even better. But this is really all there is to it. So first, you just get the slope by plugging into our slope equation, and then you just pick either one of the two points. Alright? So that's really the big idea here is that whenever you're given 2 points, you can just use either one of the two points that you're given as \(x_1\) and \(y_1\). It won't matter. Alright? So that's it for this one, folks. Let me know if you have any questions.