Welcome back everyone. This is going to be the first video in a two-part series called lines on the plane. This is the imaginative title, lines on the plane part one. The point of this video, which should be a little bit more technical than some of the other videos It to de-mystify formulas for equations of line, as you have undoubtedly seen before. The first one, which we're going to talk about in part one is what we call the point slope formula for a line. Y minus y nod = m times x minus x nod. Don't worry about all those bewildering symbols in part two we'll show that from that equation you can derive something else called the slope intercept form of the line. Y = mx + b. That's often more familiar to people, and it usually makes people more comfortable. The first one's a little bit more natural to derive. What we'll do for both of these is work up slowly, show you why the formulas do describe a line, and also tell you what it means to say that a formula describes a line. What does that weird idea even mean? Okay, let's start slowly and talk about the idea of the slope of the line segment. So, first abstractly, here's our x, y plane, here's x, here's y, Cartesian plane R2. Suppose here's the point capital A, which x-coordinate little a, y-coordinate little b. And suppose here's the point capital B, with x-coordinate little c, y-coordinate little d. Let's draw a line segment between them. Just a line between the two of them. Remember the length of this line segment is what we were calling for the distance between A and B. So the slope of the line segment between A and B, which we denote with AB with an arrow, by definition, is we usually use the letter M to symbolize it. But the definition because it's the difference between the y-coordinate of the second point, minus the y-coordinate of the first point, divided by the x-coordinate of the first point, taking away the x-coordinate of the first point, c-a. This is often seen as rise divided by run, and we'll understand by example why we use those terms. Okay let's compute a few actual real world numerical examples. So, Suppose I take the point 1, 2, there's A is 1, 2 looks about right to scale. And let's take the point 3, 3. There's the point B, it's (3,3). Let's draw the line segment between them. Let's ask that, what's the slope? So the slope of that, the slope of line segment AB, by definition, is M = 3- 2, that's the rise, difference in Y values, divided by 3- 1, the run difference in x values. That turns out to be 1 over 2, so the slope is one half. What does it mean to say that the slope of that line segment is one half? Basically that's the answer to the following question. If I were to start at a, I want to stay on the line, but I want to increase my x coordinate by a unit, how much do I need to increase or decrease my white board. It turns out here that if I start on A and I move one unit over, so there to x coordinate 2. And I want to stay on this line, what does my y coordinate have to be? Turns out it has to be 2 plus one-half. In other words the coordinates of this point 2, five-half notice the scale factor makes sense. If I go over 2 units, so I run 2 units in the x-coordinate direction, I have to rise 2 times one-half units in the y-direction, and so I get to the point 3, 3. I think that makes sense to everyone. This line segment, by the way, is what we call positive slope, because when I run, I have to actually rise up. Let me show you a line segment with negative slope. We'll take the point C here, with coordinates -1,1. And let's draw a line segment to the origin. This is 0,0, the origin. There's a nice little segment. The slope, of the line segment C0 Is 0 -1 divided by 0- (-1), careful about subtracting a negative, that's -1 over 1 is -1. So there is negative slope. That make sense, if I'm going to run one unit in the x direction I have to rise -1 or, one might say in plain English, fall one unit down the Y axis, down here. That's the idea. We've talked about the slope of little D line segments. Now, let's talk about the slope of the great big lines, in fact the equation of great big lines. So, let's take the line which goes from the point 2, 1, and go through 3,2, right there. There's a line which you can convince yourself pretty quickly. The line segment has slope one. But now let's continue that line infinitely in both directions. Down like that. Okay, and let's call this line little l. That line. Okay, here's the nice thing about it being a line, so look at the line sitting between (2,1) and (3,2), that has slope 1. If I look at the line sitting between (2,1) and any other point on the line, that has to have the same slope. So in other words, if this is some point right here on the line, x, y. The line segment between 2,1 and x,y has to have slope 1. In other words, 1 has to equal the difference of y -1 divided by x minus 2. The difference in the rise from 2,1 to x,y divided by the run. Now lets rewrite that as y- 1 = 1 index- 2. This is actually a really profound statement. Right? Because xy was arbitrary with any point on the line. This has to be true. So in other words, the line is an exclusive club which is defined by this formula that is as a set, the line is a set of 0.xy in the Cartesian plane. Such that the following relationship in x and y values is true. Y-1 is equal to 1 times x-2. Lets check if that works, right we know that three two is on the line lets ask the question is three two on the line we know it is visually but lets check if this formula works if I plug in two for y and three for x I get that 2- 1 is equal to question mark, 1 times 3 minus 2. And if you work that out this is 1 is equal to 1 times 1 and that works beautifully. So any again, if we think about sets as this little colon is like the bouncer at a club. Any point xy that wants to be on the line has to check whether this is true. So for example, the point (5,1) which is visually not on the line is also not on the line because if you plug in 5 for x into this formula and 1 for y, you get an incorrect statement. Okay, let's write that down formally what we just had. This is called a Point-Slope Formula of a line. If a line, l, has slope M and if x not y not, which usually nots to signify a specific point. Although we don't tell you what it is, is any point on the line, and l has the equation. Y- y0 = m (x- x0). Okay, in the next video we will show you a simpler formula for the line over slope intercept and we'll also work an example or two. That concludes this video.
