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Equations of Lines

In mathematics, a line is a straight line segment with zero curvature, meaning that it is a line in the Euclidean plane with no inflection points. The concept of a line is one of the most fundamental concepts in geometry, building a bridge between Euclidean geometry and algebra. In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For example, in analytic geometry a line may be defined as a set of points satisfying a linear equation, while in synthetic geometry it is described as a set of points satisfying a geometric theorem.

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Hey, guys, we're here today to talk about equations of lines such a huge topic when it comes to algebra and geometry. So we're gonna get right into it. We got a lot to talk about. There's lots of info when it comes to equations of lines. So first of all, when you think of a line, you're probably thinking of, hopefully something that looks like this to set up points that air back to back to back to go in opposite directions forever. That's the Linus. And what do we need to determine? A line? Well, two points. Determine a line. If you just had one point, then I could draw that line through it and that line through it, that line through it and so forth. So one line is not gonna be enough. That's why we need two points. So if you have two points in a plane, let's say this point and let's say this point. Then there's one and only one line that can be drawn through those two points. So first of all, uh, two points you needed, and I don't care what two points, just as long as we have two points on that line. That's all we need. But what's unique about a line? Well, a line has steepness, which we call slope, so a line convey Ari in its deepness. You can have lines that air not very steep. It all the lines that are extremely steep. You could have lines that have no steepness. This would be a horizontal line. It is no steepest. You put a ball on it, it would not move left, nor right Then you have lines that have what we would call negative steepness because it's going down to the right. That's gonna be important. Lines that go up to the right. For example, this one here, okay, which we call are classified as having positive slope because it is steeped in the upper right direction versus this line right here, which we have negative slope. This would be a example of a steep positive slope versus this one, which is a not very steep positive slope. In fact, the closer you get to the horizontal situation, the least steep you are. In fact, a horizontal line has zero slope. Since it is has no steepness, it is there's no tilt either way. You also have the special case of a vertical line. Okay, this is known as a vertical line up and down, and it has infinite steepness or, in other words, infinite slope. It is the steepest any line can be since it straight up and down. So one way of classifying lines is by its slope. It could be sloped upward to the right, downward to the right, and therefore it either has positive slope or negative slope, respectively. Or you could have a vertical horizontal line. So here's our four cases once again. Positive slope, negative slope, zero slope or infinite slow. Now, the letter that we use to define Slope is the letter M M is what we used to find slope. So if a line has a slope of M equals two versus a slope of M equals seven, this line is gonna be very steeper. Because when we get to define slope, we're gonna find out that the larger the magnitude of the slope, the steeper it iss Okay, so it's probably a good idea now to define what slope is Maybe a slide. And we talked about slopes that have values of one or two or seven pardon me. There really sitting over ones we think about it too, is the same thing is to everyone. So what this is saying is the slope of this line eyes to meaning we're going to go up to for everyone. We go over. So we're gonna go. Let's say I started here. We're gonna go up to for everyone that I go over. It's gonna look something like that. But this blind that has kind of slope of seven. That's saying we're gonna go up seven for every one that we go over 1234567 So it's gonna look like that and you can see that it's deeper than the previous one that I drew. In fact, the bigger this number gets the steeper and steeper and steeper because we're going up Mawr than we are going over The ratio of our up and down this compared to or the amount of her up and down this compared to our left and rightness has a bigger difference. So the closer we are to, let's say, zero than the least steep we are the least slope. We have the closer to a horizontal line. We are versus the larger the magnitude of our slope, the more steep it is. Well, let's say I have a line and it goes through. Let's say the point. Negative 31 And let's say the 0.24 So here's my line. Something like that. First of all, this has a positive slope because it goes up into the right. But how do you define Slope? Slope is defined by your change. We use that a little triangle is called Delta And why over the change and X So if you look at our particular example are changing Why, which is the up and down nous in this case Should be three are changing X, which is the left rightness would be 12345 This is a positive three cents up in a five to the right. So in this case, we would say our slope has a value of 3/5. So if you go so this would be unexamined the line that's and this might be subject to opinion, but 3 50 not that steep compared to something like, you know, a slope of four or eight or something larger than okay. So formally how we define slope again Changing y ever changing X If you have a point or a line part me that goes through two points and these two points are arbitrary. I'm gonna call this point x one. Why one and I'm gonna call this point down here, x two. Why, to that we can define slope is the change in why that would be why to minus y one. Or you could say why one minus y two doesn't matter as long as you're consistent. Over x two minus X one. Essentially, what we're doing is refining the difference of the wise over the difference of the excess. So if you had two points, let's say 37 and negative to four, which I could call. It's a point A and point B and I want to find the slope of a B. Alright, I'll do why minus Why again? It doesn't matter what order you do it in a long as you're consistent. So I could have done seven minus four as long as I did three minus negative two and notice something that negative in parentheses. That's also very important. If we calculate this, we're gonna get for the first one. We're gonna negative three over negative five. Or in other words, positive 3/5. And if I do the second one, I get positive. 3/3 plus two, which is five. And again we get 2 3/5. So this is an example of how you calculate slope, so every line has unique slope. So the line that goes through these two points A and B would have a slope of 3/5. Meaning if every three units upward it moves, the line goes five units to the right because it's a positive slope. Okay, how about the equation of a line? Well, if you think about our slope being equal toe y tu minus y one over x two minus x one If I multiply both sides by x two minus x one, these will cancel and I'm gonna rearranges you. Look at this in a second. Why? And then if I saw for let's say I solved for I want to solve for and make this a function of why in terms of X, then what I'm gonna do is I'm gonna solve for y two. I'm gonna add this to the other side. Or actually take that back. I'm gonna leave it as is. But what I'm gonna do is I'm gonna write it like this to make it look more like the classical definition. I'm gonna drop the sub twos, and what we have here is an equation called Point Slope. Because if you're given a point x one y one and a slope, then you can come up with an equation. So what I did is I I essentially by dropping the sub twos, Okay, in terms of notation, I'm making this. Why and this XB variable. And then this X and y pair be a very specific point that we're gonna go through. And then this is the slope that is determined by two points on that line and making this why, in this x variable, I now have a function of why, in terms of X, which is super important because if you don't have an equation of a line, I need to have an independent and dependent variable. Let's see how this works. If I have, let's say a point to common negative three, and I know that my slope is equal to, let's say, negative 1/4 using my equation Appear, I could say All right. Why? Minus my wife value equals my slope Times X minus my ex value. And I can clean this up since I have a negative negative there. This is why plus three equals negative 1/4 times X minus two and you can leave it like that. So this is called Point Slope because we're given a point, were given a slope and now we have this equation that that describes the line. So let's say I told you, I have a point at negative too three. And I know that the slope is negative. 3/4 If I plot that point Negative 23 And I know the slope is down. 3/4. 12 3/4. 1234 Then this line that I have here Okay, I can write as why minus three equals negative. 3/4 times X plus two Again, it's a plus. Here Could its a minus minus based upon the equation. So given appointments slope, we congrats the line as you saw. And now I have this equation that represents the line. And if you notice if I this is down three over four. If I go back for up three, looks like a nice point on here is negative. 66 And if our equation is right, if I plug a negative six into here, I should It should make sense that if you put it this way, if I put a negative six and for X and a six in for. Why, using this point up here, I should get truth on both sides. Six. Minus three equals negative. 34 times negative. Six plus two. Well, here we get three and we're gonna get negative 3/4 times. It looks to be negative. Four. The force will cancel and there's a negative left over. So it's negative. Three times negative one you get three equals three. That should make sense, because this is the equation that represents this line in its entirety. Okay, so that's an example of Point Slope and how to use points. And if you're given an equation of a line, let's say, let's say I had a picture of a line, not the equation. I had a picture of a line that did something like this. Let's say I had a line that we went from here to here to here. Okay, so there's my line right there and I said, Let's find an equation of it. Well, you can pick any point you want, as long as it's a nice point that's on that curve. So I'm gonna pick, Let's say this point right here and you can tell from the picture that the slope is up 1/2 up, 1/2. So this point right here is negative. Three. Negative three. So I can say, All right, my slope is a half, and the equation is why plus three equals one half Times X plus three. But I could have also picked let's say this point, which is the point one negative one. So another equation I could have written is why plus one equals one half times X minus one. Now, you know what you're saying is like saying, Wait a minute. These air two different equations for the same line. But here's the thing. There's an infinite number of points on online, so you can pick an infinite number of of points that you know, a focus of interest. However, every line has unique slope, so even though these look different Algebraic Lee, they do represent the same line. So more of the story here is there's an infinite number of points slope forms for any given line, but we're gonna find out when we convert this to the next form, which is called Slope Intercept. There's one and only one of those types of equations because every line has unique slope, and every line will have a unique place where it crosses the Y axis. So if we have a line in Point Slope, here's an example why minus three equals, Let's say two times X plus four. If I want to solve directly for why and get why, all by itself, I might do something like this. Why minus three equals two X plus eight Or, in other words, y equals two X plus 11. What I've done is I've converted this point slope equation into what we call Slope intercept, and here's the reason why it's called Slope Intercept. We still maintain the slope, but this number right here ends up being the Y intercept. Why is that? Well, if you plug to zero in for X, you get y equals two times zero plus 11, which is just why equals 11, which means the 0.0 11 would be the Y intercept. And we know it's the Y intercept because of the zero here. So anything in this form, which is in general, why equals m X plus B okay is known as the slope intercept equation. And each version, like Point Slope versus Slope intercept just has different pros and cons and meaning it's gonna give you different things about the line. So if you go, you graph a line and I have y equals, let's say one half X plus three. I know that I can start on my Y axis at three. And then the slope is gonna tell me to go up one over to, and now I can plot that point. And now I can grab my line. So the benefit of slope intercept is you know exactly where it crosses the Y axis. And then from there you can use your slope to proceed. So that's example of slope intercept and notice. I started with Point Slope, and I converted to this thing which we call Slope intercept. Okay, so there's one other form of the line, and that is this thing called standard form and standard form of a line again, different pros and cons. Standard form looks like this where A and B and C are integers and a is positive. So if I had something like y equals negative three x plus seven inches slope intercept. If I added the three X over to the other side, I would now be in standard form. Or if I started with something like why minus two equals one half times X minus eight, which is in Point Slope. I could first convert to slope intercept, then bringing the one half X over. I'd have negative one half X plus y equals negative two. Now this is not a standard form for two reasons. One are a value, which is negative. One half is not a it's a fraction, and it's not positive. So if I multiply both sides by negative too, then we get positive. X minus two y equals positive for and now I have converted a standard form, and what's nice about standard form is that if so, the other forms give you a sloping appoint. One of them gives you a very specific point, which is the Y intercept. The standard form is going to give us our X and Y intercept. If you recall an X intercept is when. Why is zero and the Y intercept is when X equals zero. So if you plug in white equals zero, think about it. If you're plugging Michael zero, the whole thing goes away. You're left automatically with X equals four. So now you know your X intercept. If you plug in a zero in for X, you're left with negative two y equals for in other words, why equals negative too. So now you know, zero negative too. Is your y intercept And what do we say we know is true? But are we need for a unique line? I need two points. Well, now I have two points. I have 1234 comma zero And I have negative as your negative too. And there you go. Now I can draw my line. And then also indirectly, we have our slope, so it looks like our slope is up to over four. In other words, it looks like our slope is a half. Where does that show up in our in our equation. Do you see how there's a one here and there to here? Given a X plus B y equals C, your slope will always be negative. A over B. So if you want to use that shortcut, you can generate your slope fairly quickly along with your X and Y intercepts. So this is just kind of a recap recap, but intro to what an equation of the line is. What slope is and the three different forms, which is Point Slope, Slope intercept and your standard form, I'll be back with some examples.