Summary: Forms of two-variable linear equations
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- [Instructor] We've got the equation Y plus two is equal to negative two, times X minus three. And, what I wanna do is figure out what is the slope of the line that this equation describes? And there's a couple of ways that you can approach it. What my brain wants to do is well, I know a few forms where it's easy to pick out the slope. For example, if I can manipulate that equation to be in the form Y is equal to MX plus B, well then I know that this M here, the coefficient on the X term, well that's going to be my slope. And B is going to be my Y intercept, we cover that in many other videos. Another option is to get into point-slope form. So the general framework or the general template for point-slope form is, if I have an equation of the form Y minus Y1 is equal to M times X minus X1, well then I immediately know that the line that this equation describes is going to have a slope of M once again. And here the Y intercept doesn't jump out at you. Let me make sure you can read this over here. The Y intercept doesn't jump out at you, but you know a point that is on this line. In particular, you know that the point X1, Y1 is going to be on this line. X1, Y1. So let's look at our original example. So it might immediately jump out at you that this is actually in point-slope form. You might say, well okay, I see I have a minus X1, so X1 would be three, I have my slope here and that answers our question, our slope would be negative two. But here it says plus two, I have to subtract a Y1. Well, you could just rewrite this, so it says, so you have Y minus negative two is equal to negative two times X minus three, and then you see it's exactly this point-slope form right over here. So our slope right over there is negative two, and then if I were to ask you, well give me a point that sits on this line, you could say, alright, an X1 would be three, and a Y1 would be negative two. This point sits on the line, it's not the Y intercept, but it's a point on the line and we know the slope is negative two. Now another way to approach this is to just manipulate it so that we get into slope-intercept form. So let's do that, let's manipulate it so we get into slope-intercept form. So the first thing my brain wants to do is distribute this negative two, and if I do that, I get Y plus two is equal to negative 2x, negative two times negative three, plus six. And then I can subtract two from both sides, and then I get Y is equal to negative 2x plus four. And so here I am in slope Y intercept form, and once again, I could say, alright my M here, the coefficient on the X term, is my slope. So my slope is negative two. Let's do another example. So here, this equation doesn't immediately go into either one of these forms, so let's manipulate it. And if it's in either one of them, I like to get into slope Y intercept form, it's a little bit easier for my brain to understand. So let's do that. So let us collect, well let's get the Xs, let's just isolate the Y on the right-hand side, since the 2y is already there. So let's add three to both sides. I'm just trying to get rid of this negative three. So if we add three to both sides, on the left-hand side we have negative 4x plus 10 is equal to 2y, these cancel out. That was the whole point. And now to solve for Y, we just have to divide both sides by two. So if we divide everything by two, we get negative 2x plus five is equal to Y. So this is in slope-intercept form. I just have the Y on the right-hand side instead of the left-hand side. We have Y is equal to mx plus B, and so our M is the coefficient on the X term right over here. So our slope is once again, is negative two, and here our Y intercept is five, in case we wanted to know it. Let's do one more example, one more example. Alright, so once again, this is in neither slope-intercept or point-slope form to begin with, so let's just try to get it to slope-intercept form. And like always, pause the video, and see if you can figure it out yourself. Alright, so let's get all the Ys on the left-hand side isolated, and the Xs on the right-hand side. So let me get rid of this negative 3x. So I'm gonna add 3x to both sides, and let's get rid of this 3y right over here. So let's subtract 3y from both sides. You couldn't do this, I'm doing two steps at once, but once again, I'm trying to get rid of this 3x, so I'm trying to get rid of this negative 3x, so I add 3x to the left, but I have to do it to the right if I want to maintain the equality. And if I want to get rid of this 3y, well I subtract 3y from here, but I have to do it on the left-hand side if I want to maintain the equality. So what do I get? That cancels out, 5y minus 3y is 2y, is equal to 2x plus 3x is 5x, and then these two characters cancel out. And so if I want to solve for Y, I just divide both sides by two and I get Y is equal to five halves X, and I'm done. And you mights say wait, if this doesn't look exactly like slope-intercept form, where is my B? Well your B, if you wanted to see it, you could just write plus zero, B is implicitly zero right over here. So your slope, your slope is going to be the coefficient on the X term, it's going to be five halves. And if you want to know your Y intercept, well it's zero, when X is zero, Y is zero.