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Equation of a Line - Overview

In mathematics, a line is a straight line segment with zero curvature. The most familiar types of lines are the lines in Euclidean geometry, which are described by two numerical parameters, and the lines in non-Euclidean geometry, which are described by more than two parameters.


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Okay, Okay. So in this video, we're gonna be talking about equations of a line. So let's go and start out by talking about what a line is. So a line as it might sound in modern English, it's just something that's straight that has at least two points aan den. You can connect those two points to make a line. So think about the line is a lot of people think that anything that can be drawn is a line. So some people think like a parabola. The line, something like this is a line. This is a line, but we actually know that that's actually not true. The only thing that's a line is a straight, um, lying. So something like this one right here. So then, in that case, let's think about what the options are for, you know, writing out the line. So we already talked about how we need at least two points to make up a line. Okay, But we also know that there are some equations that we can use to find what a line actually is. So the very first one and this is probably gonna be your most important equation. The one that probably comes in the most handy. So that's gonna be your why is equal to m X Plus it be. And this one is called the Slope intercept formula. Okay? And the reason why it's called the slope intercept formula is because of the fact that we have a slope right here and an intercept right here so that M stands for slope. It's kind of denote slope that be value is our y intercept? Yeah, well, then let's go in and talk about what those two vocab words even mean. Well, slow basically just means how steep So how steep the lioness and our Y intercept is basically just a point where the graph two point where the graph intersex um, the y axis. Look, I know that's kind of a lot of vocab, so let's go ahead and put that on an actual graphs. We can visualize it a little bit better, so I'm gonna draw out of graph here. So here's my X axis, and here's my y axis, because this right here, that vertical line is R y axis. If we had a line that looks something like this, then this point right here is where it crosses that vertical. Why access? And so that point the orange point right there is going to be RB value because that is where the graph intersects the Y axis. So then the slope is just how steep it is. So if we have a slope that something like this going through the same be point, then that's gonna be a little bit steeper than the orange. And then if we have one that's a little bit less steep, it probably looks something like this. So the blue one has the smallest slope, and then orange has a bigger slope. But then the green has the biggest slope because it's the steepest. Hm. So then, in terms of the slope, well, there's a couple more things that we have to know. So just from basic pre ology. But we know that this is our first quadrant. This is our second quadrant. This is our third quadrant, and here's our fourth quadrant in the first quadrant. We know that both the X and the Y values are positive. The second one, um, the X is positive, but the why is for the X is excuse me negative and the why is still positive because we know that the why is the vertical measure and that's still gonna be positive. Whereas the horizontal measure is gonna be negative. So then for quadrant three, we know that both of them are going to be negative, because in this area, not only is your ex negative, but your wife is also negative. And then for quadrant four, we know that the excess positive, but the why is gonna be negative. So then, in terms of our slope, we know that there's going to be a negative slope when it's going downwards. So all those three lines I have drawn currently are going up. But if I have a slope line that looks something like this, well, that one is going down. It's facing down. It's starting from a higher point and ending at a lower point. And so that one has a negative slope. Okay, so that's how to tall, Whether you have a positive slope or negative slope, if you have an equation, obviously you could just look at that end value. And if that end value is a positive number, that's gonna be a positive slope. Whereas if that end value is negative. That's gonna be a negative slope. That's pretty easy. So then let's think about how we can find the slope. Well, the slope. There's a couple different ways of finding it. If you don't know you need either a point. You need other two points to find it. So, for instance, you can have two points. Need to points. Do you find the slope? So let's go toe look at an example. So if we have the points to common four and six Kama 10. Let's try to find the slope between those. Well, that means that we need toe bring out an equation toe relate that. So the equation for Slope is just the change. And why over the change and ex that triangle just means change. From here, though, we know that that basically just means that we could take the second white point and subtracted from the first white point. So why? To minus y one over x two, minus X one. So then from here, if we just plugged in those points, here's let's just call this one our second point, and this one is our first point so that our second why? Value is 10 our first y values four. So 10 minus four has changed in our wise and then our second X value is six and our first X value is too. So that's our change in X is so then if we do 10 minus four, that gives us six six minus two gives us four. So if we simplify that down, we get three over to. So we have just found. But the slope between these two points would be three over to. So that's basically how you find the slope. So again, really, all it is is you just have to find the change in your wise and the change in your exes, So that's not too complicated. So then let's think about Well okay, well, if you have those two points Well, how do we find the, um, the intercept? Because we've already dealt with what it means to have the slope, but we're gonna go ahead and look at what it means to find the intercept. Okay, so let's go ahead and bring out those same exact two points. So it's gonna be, uh oh, I'll go ahead and do so. We're going to say the points to calm before and then six. Comic 10. So the same exact points and we already found that are slope was 3/2. So I'm going to say I'm is equal to three over to Well, then at this point from our Y is equal to M X plus B equation. We already know RM. So I'm just gonna plug that in just so that it gets rid of one extra variable. Less variables we have, the easier it is. So I'm going to say why is equal to 3/2 x plus B? So then, at this point, in order to find B well, you do have a couple of points here, so you just have to plug in either this point or this point. It really doesn't matter into your ex and why? So I'm just going to stick with our first point because that looks a little bit easier. So then I'm going to get 3/2 times are X value, which is the to plus R B value, which is why intercept should equal for so then if I basically try to solve this out while 3/2 times to the two on the top of the two on the bottom councils out, and that just leaves us with three. So four is equal to three plus RB value. So again, in order to isolate RB value, we just have to subtract three from both sides. And then we get that one is equal to R. B. So that's how we found there. Be value. So in order to find RB value, we know that we just have to plug in the slope once you find it and then plug in a point. So basically, all you really need in order to find the equation of a line is either two points or a point on a slope. So two points or point and slope. And the reason why I say point and slope is because if you have to point, you can find a slope, and at that point you can use that slope to find and appoints to plug it in so you don't necessarily have to have two points. You can have either of two points or a point in a slope. Mhm. So then we've discussed how we find the equation of a Y is equal to M X plus B line, which again is our, um, slope intercept formula. So that's pretty important there. So then let's talk about Well, how do we find the equation of a different type of lying? So the next thing that we're gonna discuss is the standard form. So standard form is just going to be, um, why is equal to X plus B? Why is equal to see and that's actually gonna equals zero. So zero is equal to X plus B Y, um, plus C So some people will write as this or some people will write as a X plus B y equals C. So C is just a constant, so that doesn't really matter too much. So on the S a T. They'll never really ask you to do anything with the Y intercept for standard form. Typically, what they'll ask you to do is find the slope. Mhm. Then, if you're looking for the slope of a standard line, we know that the equation for that is just going to be negative. A over B. So here's the thing is that a lot of questions will have, like four different equations in standard form for the answer choices and one equation in maybe why it goes, I'm exposed to be formed this slope intercept form from the question and they'll ask you. Okay, well, which of these have the same two slips? So many people are not very familiar with our standard form that they'll tick, tick all those four standard form equations and convert that into y equals MX plus B. But you can imagine that's gonna burn a lot of time. So that's not a very ideal way of solving the question, because not only is it a lot of time, but by doing all that conversions, you're just giving yourself a lot of room to make small errors such as like maybe forgetting to switch a negative sign, thinking that it, too, is a three actually multiplying dividing audience instead of subtracting those air all small areas that you're allowing yourself to make. So instead of converting those entire equations from standard form to slope intercept, I would just be familiar with finding the slope by using em is equals negative. A over B came. So then let's go ahead and look at one. So, for example, if we have three X minus two Y is equal to 10. That's totally valid Standard form equation. So again, in order to find our slope, we just have to do the negative a over B. Well, our a by value is whatever is associated with three X, so that's gonna be a negative three. And then RB value is gonna be whatever is associated with a y. So that's gonna be the negative tomb I do that the negatives on the top and bottom will cancel out. I'm going to be left with three over to. So then the slope of this line right here, three x minus two y is equal to attend is going to be three over to. So that's how we did that. So then the one thing that you have to be very careful of is if the equation is not written in a very nice form. So, for example, they might have something like eight y plus seven x negative. Eight y plus seven X is equal to 12, and they'll say, Okay, so what is the slope of that? And so many people, without thinking, we'll just say Okay, well, it's just negative. Ever be so I'm just gonna take negative of negative eight and divide that by seven. But we actually know that that does not work. And you are making a huge mistake there because remember, are a value is whatever is associated to the X, not the thing that comes first. It's whatever is attached to the X, so the A value should actually be seven. So it be negative seven over the B value, which is gonna be negative. Eight. So the slope is actually gonna be 7/8 as opposed to 8/7. And if you get one thing wrong with the slope, then that's going to get the rest of your question wrong to just be very, very careful with fat. Thank So those are basically the two equations that you really have to work with. There is a third equation called the point slope, but that one is not really used very much in the S A. T. So there's not really a point in, um, really learning it. You don't really need to be that familiar with it. Why equals MX supposedly is the really main one. I would say that accounts for about 80% of the questions, and the standard form does account for about 30% of the questions. So definitely is still good to know it. But the point slope rarely, rarely, rarely comes up, and so you don't actually have to be that aware of that one.

Johns Hopkins University
Top SAT Educators
Emma C.

Wellesley College

Martha R.

Michigan State University

Nick J.

University of Wisconsin - Milwaukee

Nooshin S.

Other Schools

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