Everyone. So as we've talked about equations with two variables, we've seen different shapes of graphs. So for example, we've seen perfectly straight lines, but we've also seen things that curve in different ways. Well, for the next few videos, we're only just gonna be focusing on lines. And we're gonna start our discussion by talking about a very important characteristic of lines, which is called the slope. I'm gonna show you in this video is that the slope is really just a number that you can calculate by using any two points on a line. It's very straightforward. Let me go ahead and take a look. Let's go ahead and take a look here. So the slope is really just a number that represents how steep a line is. So take a look at these two lines and imagine they were like the side of a hill that you're trying to climb. The green one is a little bit more shallow because you don't have to rise as much or go up before you can go over. Whereas the red line is a little bit steeper, it's like a little bit more vertical. This would be a harder climb mathematically what's going on here is it's how much the Y value changes divided by how much X value changes. So let's take a look at the equation here. The variable that we use for uh the slope is the variable M and the equation is Y two minus Y one over X two minus X one. So I'm gonna talk about what those things mean in just a second. But this is a little bit tedious to write out. So we've developed a little bit of a shorthand for this and we use a little triangle symbol which is the Greek letter delta. The way that this is written is delta Y over delta X just to make it a little bit more sort of short. And basically what the delta Y means is, it's kind of like the rise of an equation or a graph divided by delta X which is like the run again. Think about the steepness here and think about these two lines. Imagine I was trying to get from this point to this point. First on this graph graph, I have to rise a little bit, I have to go up vertically before I can go over. That's what those delta symbols mean here. All right. So anytime you see this triangle symbol that just means a change in a variable or a difference and based what's going on here with these Y twos and Y ones and all that stuff is these are actually just points. So remember that we can locate any point in a graph which is by an XY pair. But since we have two of them, we just give them numbers. And so really X two and Y two are just an X one and Y one are just two points that are on a graph. And they may be given to you or you might actually have to pick them yourself. Let's go and take a look at an equation or, or an example. So I can show you how this works. We're gonna find the slopes of these two lines that we've been seeing in this graph. So basically, we have to do is to find the slope of A or the, yeah, the slope of A, I just need Y two minus Y one over X two minus X one. So what are those two points over here? Well, here on the graph they're actually given to us, we just have to figure out what those ordered pairs are. What are those points? X one, Y one. So this point over here, I'm gonna use this X one, Y one and this is just the 0.1 comma two, right? So this is one, comma two. And what's my X two, Y two? It's just two and then four and the Y. So this two comma four. So these are my two ordered pairs and I could basically just plug them into this equation over here. And then just calculate. So Y two is the Y coordinate of the second point which is four minus Y one, which is two and then X two minus X one is the X coordinate of two minus the X coordinate of one. So that's two minus one. So what happens is you just get 2/1. And so therefore, the slope of A is just equal to two, right? Another way you can kind of think about this two or there's 2/1 is what's the rise over the run from this point to this point? How much did I rise by? I went from 2 to 4. So my rise is two and then what's the run? I went from two or sorry, I went from one over to two over here. So that means the run is just one. So in other words, the rise over the run was just 2/1. And that's why we got a slope of two. Let's do the exact same thing. Now, for points for line B, all right. So the slope is two. So for line B, what we're gonna do here is we're, we're, we're gonna get an X one, Y one or an X two Y two. All right. Now, in this case, the actual points aren't given to us already. So we're gonna have to go pick them. And basically what you want to do here is you always want to pick points that are sort of like sit at the uh sort of intersection of your grid. You don't want points that are kind of like halfway in between a point because it's really hard to tell what numbers are. So these are bad points. You're is gonna look for points like this over here or this over here. You can, where you can very easily tell what those coordinates are. All right. So in this case, this is gonna be my X one Y one and that's gonna be my X two Y two. All right. So what are those points? Well, this point over here is really just four common negative one. And this point over here is really just uh five comma four. All right. So my X one Y one is four comma negative one and this is just gonna be five comma four over here. So let's use our equation now for the slope. So my MB is really just Y two minus Y one over X two minus X one or it's really just delta Y over delta X. So let's try to use the rise over the run sort of technique. Uh What's the rise in this equation? How much do I have to go up by before I go over? Well, look at this point, how much do I have to go up? I have to go 1234 and five. So in other words, the delta Y over here is five and what's the run? Well, I'm really just going from 4 to 5. So the delta X is just one. All right. So, really, without having to plug in all these points, you could just figure out that it's just 5/1 and then the slope of this line is just equal to five. All right. So let's take a look at these two numbers. Now, you got two and then we got five for the red line. So notice how I said earlier that the green line would be easier because it's a little bit shallower, right? It's a little bit of an easier climb because it's not a steep. Whereas the red line is way more vertical. Basically, what we can see here is that shallower lines have lower slopes. Um And steeper lines have higher slopes. So a slope of five means you have to go up five before you can over, go over one. All right. Now, one last point of to make here is actually, uh, you know, we picked these as our X one, Y one and X two Y two. But the order of points actually doesn't matter because it won't affect what your slope is. Generally. What's best to do though is set your X one Y one as the leftmost points. Uh Basically, just because it's gonna introduce the fewest amount of negative sign. Let me just show you what I mean by this. Where did you do this last example, we're basically gonna recalculate this slope over here for a but we're gonna reverse the points. So now our X one, Y one is gonna be two comma four and then our X two, Y two is gonna be one comma two. But we're gonna see that the math works just the same way. So our ma is just gonna be Y two minus Y one over X two minus X one. And really just what happens is uh if we're just gonna use Y two minus Y one, now this becomes two minus four divided by X two minus X one. That's one minus two. So one minus two. All right. And we're gonna see here that happens is we're gonna get negative two over negative one. The negatives actually cancel and you still just end up with a slope of two. So reversing the points actually doesn't affect the slope, but you just get a few more negative signs and that's a little bit harder. All right. So that's it for this one, folks. Thanks for watching.

2

Problem

Problem

Find the slope of the line shown below.

A

$m=1$

B

$m=\frac23$

C

$m=\frac32$

D

$m=3$

3

Problem

Problem

Find the slope of the line containing the points $\left(-1,1\right)$ and $\left(4,3\right)$.

A

$m=\frac52$

B

$m=\frac25$

C

$m=2$

D

$m=\frac43$

4

concept

Types of Slope

Video duration:

5m

Play a video:

Everyone. So in a previous video, we were introduced to slopes, which is the idea of the steepness of the line or a hill or to the rise of the run. Mathematically, it was delta Y over delta X. And up until now, all the slopes that we've seen have been positive. I'm gonna show you this video is that when you calculate slopes, it actually can fall in one to, in one of four categories, it could be positive or could be negative numbers. You can get zero slope or you can get this new thing which is called undefined. And I want to talk about each one of those things and show you the differences. So let's go ahead and take a look here. All right. So let's take a look at the line that we've already sort of seen before. Something that looks like this and let's calculate the slope really quickly. So remember I just need two points. Any two points works. It doesn't matter which ones I pick. So I'm just gonna pick this one over here and this one. Remember, I can't use lines, I can't use points like this because this is like halfway in between a number and that's bad. So if I want to calculate the slope, the rise over the run between these two points, I just have to figure out how much I have to go up and then over. So my delta Y, so how much I have to go up is really just two and then how I, I have to go over by one. So my delta wise two, my delta X is one. And so I just the slope of two. All right. So we've seen that before. Now, let's take a look at the red line because the red line looks different. It doesn't look like this. It actually looks like it's this. So why does that look like that? Well, let's do the same thing and calculate the slope. I can just pick any two points. It doesn't matter which ones I pick. And it turns out that these are actually all sort of fall on nice sort of intersections of these tool grids. So I can just pick any two points, it doesn't matter. So what do I have to do to get from this point now to this point over here? Well, in this case, what happens is if I'm moving to the right, I actually don't have to go up, I have to go down in order to get to the next point instead of moving up like I did for this blue line, I have to go down. So it's not like a rise, it's kind of like a fall, right? And so mathematically what happens is that instead of a delta Y being positive, the delta Y is gonna be negative, you're gonna have negative one, you have to fall one and then you have to go over by one. So what's the delta Y over delta X? Well, here it's just gonna be negative one over positive one, which is negative one. So clearly we can see here that the reason these two lines are different is because the signs of their slopes are different. So basically what happens is if you have a line that goes up from left to right, like our blue line over here, the slope is positive. And if you have a line that goes down from left to right, like this one over here, then the slope is gonna be negative. All right. So that's really all there is to it. So you have positive slopes looks like this negative slopes and it looks like this. That that's the difference. So let's take a look at a couple of other situations here. So let's take a look at this purple line. This purple line is perfectly and horizontal. So let's calculate the slope of this one. Well, if I pick any two points again, it doesn't matter uh which points you pick, let's just pick this one. And this one over here, I want to calculate the delta Y over delta X, what's the rise over the run of this line? Well, if you look at this, the rise is actually just nothing. I don't have to rise at all because it's perfectly flat. In other words, the change in the Y value is zero, right? The Y value is two. And then over here it's two again. So I haven't changed anything. So delta Y is two sorry, delta Y zero, but the delta X is the change in the X and I've just gone over two in this situation. So here my delta wire, delta X is zero over two and it actually doesn't matter what you plug in on the bottom of this equation because the slope will always be zero. So remember the slope is the steepness of the line and we got zero. And that makes perfect sense because a steepness of zero would just be a perfectly flat hill. Um And that's why the slope is zero. OK. So let's take a look at this vertical line here because the situation is a little bit uh sort of similar. Um So let's take of any two points. I'm gonna pick this point over here and this point again, I'm just picking them at random, doesn't matter which ones you pick. So what's the rise of the one run? Well, well, in order to get from this point to this point, I have to rise by three, that's my delta Y. So what about delta X, how much did you run over to get to the next point? Well, actually you didn't because again, we were here at uh X equals two and then again, we're at X equals two. So the delta X is just equal to zero. So what this looks like is that your rise over the run is gonna be three divided by zero. And this is one of these weird math things that you can't do. We've talked about this before. You can never divide by zero. This is bad. So what we do here is we say that the slope of this line is undefined. It's a number that you can't really, it's actually not really a number you can't define this because you can't divide by zero. All right. So vertical lines, the slopes are undefined. Whereas for horizontal lines, it's perfectly fine. That just means that your slope is equal to zero. All right. So the last thing you have to know about these equations of these lines is their equations. Um Basically what happens is that when you have a horizontal line, it's gonna look like Y equals some number. Like for example, this perfectly flat horizontal line that we have here because all of the Y values are always two. This equation is just Y equals two, no matter what the value of X is Y will always just be two no matter what. So the general form of this equation is just gonna be Y equals some number, sometimes textbooks, we use the letter B uh for this. And then for vertical lines, it's very similar for this line, the X value is always equals two no matter what the Y value is. So, so vertical lines will always have X equals some number. And I think usually textbooks will use a letter A for this. All right. So that's gonna be important to know just knowing those two sort of forms of equations anyway. So, thanks for watching. That's for this one.

5

Problem

Problem

Graph a line with a slope of 0 that passes through the point $\left(3,-2\right)$.

A

B

C

D

6

Problem

Problem

Which of the following graphs below represents the equation $x=3$?

A

B

C

D

7

concept

Slope-Intercept Form

Video duration:

3m

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Everyone. So throughout our discussion lines, we've seen problems with that ask us to calculate the slope of a line by looking at the rise of the run between two points. But some problems like the one we, we're gonna work out down below won't ask for just that some problems will ask us to look at the graph of a line and write its equation in a very specific way called the slope intercept form. So that's what I want to talk about in this video. And what I'm gonna show you is it's basically just a very specific way that we write an equation of two variables using two things we've already seen before independently. So we can use its slope and its intercept uh specifically the Y intercept. All right. So I'm gonna show you this equation and it's probably something that you've seen before. So let's get started. So the slope intercept form of an equation is actually just an equation that's Y equals MX plus B. You've probably heard that at some point in a math class Y equals MX plus B. This is just the slope intercept form of a line. It's one of the most easy and straightforward ways to describe a line equation. So let's get started in talk about these two variables here. The M is basically just the slope. And we've already seen that before we calculate that by using delta Y or delta X rise of run uh or you can just use the sort of longer format here. So in this equation, what we can see is that the rise of the run between these two points is 2/1. And so the slope is just two. What about the Y intercept? Well, the B term over here is the Y intercept. And we've already talked about that separately. Y intercept is basically just the Y value wherever the graph crosses the Y axis, it's where X is equal to zero. So for instance, in this case, the graph crosses the Y axis right over here, the Y value at this point is three. So that is the B term, right? So that's Y intercepts. So to put these things together, Y equals MX plus B, all you have to do is the equation here has a slope of two. So that goes in front of the X and then the intercept is three. So the equation of this line is just Y equals two X plus three. That's how to describe this equation in slope intercept form. That's really all there is to it, all right. So that's all there is to slope intercept form. Let's go ahead and take a look at another example. All right. So in this graph below, we're gonna identify the slope inter or sorry, the slope and the Y intercept. And then we're gonna write the equation in slope intercept form. So remember slope intercept form here is just gonna be Y equals MX plus B. In order to figure out M and B, we're gonna have to take a look at the graph. All right. So the al always the easiest sort of value to start with is gonna be the B term because you really just have to look at the graph and figure out where does it cross the Y axis? You don't have to calculate anything. So let's take a look at this graph where does it cross the Y axis? Well, it crosses right over here. So the, what's the Y value where this graph crosses the Y axis? It's just negative three. So that's the B, it's just negative three. That's all there is to it. You don't have to write the ordered pair. It's just the Y value. All right. So how do we calculate M? Well, for M we're gonna have to use rise over run. So we're m, we're just gonna calculate this by using delta Y over delta X or if we're given two points, you could just plug those points in. But basically, if I'm gonna have this point over here, what's the rise of the run to get to the next points. Well, if I take a look at this graph here, uh what happens is I've got all these different points that are all at the intersections of these little lines here. All these are valid points that I can pick. So what's the rise of the run? Well, to get from this point to the next one, all I have to do is I have to go up one and then over one. So up 1/1. So that's just a rise over run of just one. So the slope is one, I just go one over and uh sorry, one up and one over. So because the slope is just equal to one, I basically just plug both of these things into this equation. And my equation is just equal to Y equals one X plus. And this is gonna be negative three just to simplify this. A lot of times that you'll see here is if there's just a one in front, you don't even write it. So it's just Y equals X and this is gonna be minus three. So this is the equation that describes this line in slope intercept form Y equals X minus three. All right. So anyway, folks, thanks for watching. Let me know if you have any questions.

8

Problem

Problem

In the graph shown, identify the y–intercept& slope. Write the equation of this line in Slope-Intercept form.

A

$y=\frac23x+1$

B

$y=-\frac23x+1$

C

$y=-2x+1$

D

$y=x+2$

9

concept

Graphing Lines in Slope-Intercept Form

Video duration:

2m

Play a video:

Everyone. So in a previous video, we were introduced to the slope intercept form of a line Y equals MX plus B. And we saw some problems where a graph of a line was already given to us. And we wanted to write the equation in slope intercept form. But in some problems like the one we're gonna work out here, you actually will help do the opposite. You'll have the equation that's already given to you and you'll be asked to graph it. So that's what I want to show you how to do in this video, how to graph lines when you're given the equations in slope intercept form. And basically what I'm gonna show you is that a line equation in this form tells you everything that you need to grab it very quickly. And I'm gonna show you a step by step way to do that. So let's just jump right into our example and get started here. So we have the equation Y equals two thirds X plus one. And the first thing we wanna do is we wanna identify the Y intercepts and then the slope over here. So let's get started. Remember the Y intercept is just the B term in Y equals MX plus B. So if you look at the equation, why equal two thirds X plus one, which you'll see here is the B is basically just the one, it's the thing, the constant at the end and the M is just the slope. It's the thing that goes in front of the X which is two thirds. So right from this equation, we could just immediately pull out the fact that the B term is just one that's the Y intercept and the slope is just a fraction which is two thirds. So how do we graph this? Well, the first thing you're gonna do is you're just gonna plot the Y intercept. That's the easiest thing to plot because you don't have to calculate anything. You're just sort of marking a place on the graph. The Y intercept is the Y value where it crosses the Y axis. So I know that this graph is gonna cross through this point, but that's not enough information to graph this because I don't know if it's gonna look like this or this or that. So I'm gonna need more points. So that actually brings us to the second step here, which is we're gonna plot at least one more additional point. And the way we do that is by using the definition of slope, remember, slope is just rise over run. So in other words, we have to go rise two and then we have to run over by three in order to get to the next point. So basically, you're gonna take your y intercept here and you're just gonna go up two, you're gonna rise two and then you have to go over three, just do rise over run to get to the next point. You can either go up to the right or you could go down to the left. All you really need is just one additional point. Now, once you've done that, now you're basically done because all you have to do is just connect the points with a line. So again, you could have gone downwards like this and this basically would have just gone down two and over three. But anyways, you would actually end up seeing that the equation of your line kind of goes through all three of these points and it looks something like this. All right. So that's how we go from an equation to a graph in three simple steps. Hopefully, that makes sense. Thanks for watching.

10

Problem

Problem

Identify the 𝒚–intercept& slopeof $y=-2x-3$ . Then graph the equation.

A

$b=-2,m=-3$

B

$b=-3,m=-2$

C

$b=\frac23,m=-3$

D

$b=2,m=-3$

11

concept

Point-Slope Form

Video duration:

5m

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Everyone. So in earlier videos, we saw the slope intercept form of a line Y equals MX plus B. And usually we use this form whenever we were given information about the slope and intercept and asked for the graph of the equation or the other way around. So for example, we were asked to graph something like why equals two thirds X minus one. But I'm gonna show you in this video is that sometimes you might be given an entirely different set of information, you might be told something like a slope and then it passes through some points. So when you're asked to write an equation of a line that passes through a point that is not the Y intercept, I'm gonna show you that we actually use a different form of writing an equation called the point slope form. And I'm gonna show you the, the differences and similarities between the two equations. So let's go ahead and get started here. All right. So again, I, I first want to talk about when you use these two different forms, we use Y equals MX plus B whenever we're given information about M and B and ask for the graph or the other way around. So for example, two thirds X minus one, we're given the M and the B and we're asked to graph it and we can see how to do that very quickly here. We just know that it's going to pass through the 0.0 comma negative one, that's the Y intercept. And then we just sort of graph the next point by using rise over run, the slope is two thirds, you go up to over three. And basically this line is gonna look something that looks like this. All right. Now, let's take a look at the equation or the example that we're gonna figure out uh the solve here, which tells us that we're gonna have to write the equation of a line that has a slope of two thirds and it passes through some random 0.3 comma one. So basically, whenever you're given information like this, whenever you're given the slope like we have here and some random points, which is gonna be something like X one Y one, then you're gonna use this new form called the point slope form. You also could be given two points of information like X one, Y one and X two, Y two. All right. So that's the times where you're gonna use this new point slope form. And basically, it's Y minus Y one equals a parentheses X minus X one. So really with this equation, there's three numbers that you need to plug into this equation because these Ys and Xs actually don't get replaced with numbers. All right. And so the reason we call it a point slope is because X one and Y one really are just the point that it's giving you. So let's get started here with part A. We're going to write the equation in this new form. So I'm gonna take this equation here. Y minus Y one goes m parentheses X minus X one. So I first need the slope of the line. And so I'm gonna go ahead and figure that out first. And we actually already told directly what the slope of this line is. It's just two thirds. So we already have what that number is. So now all we need to do is just figure out the X one and the Y one. So what is that? Well, really, it's just the point that they're telling you that it passes through when they say that it passes through the 0.3 comma one. What they're saying is that this line is gonna pass through this point on the graph over here. That's just a coordinate. It's just, they really just giving you the X one and Y one that you plug into the equation. So it's really straightforward, you just take these numbers and just pop them right into this equation over here and you're basically done. So if I wanna just rewrite this equation, this is gonna be Y minus the Y one coordinate is one that equals and this is gonna be the slope which again, we know the slope is just two thirds. So two thirds over here and this is gonna be X minus X one is the three. All right. So this is the equation in point slope form of the line that passes through this point over here and has a slope of two thirds. That's it, That's the point slope form. All right. So we're basically done with part A. Now, in part B, we're gonna graph the line. So how do we do this? Well, again, we know this line is gonna pass through this 0.3 comma one, but that's not enough for me to build the line because there's only one point. So the way we did this for slope intercept form was once we got the Y intercept, we got the next point by using the slope. It's a very similar idea here. We know it passes through this point over here. So I can get the next point by using rise over, run in the slope. So I can go up two and over three and I'll end up over here or I could go down to the left. And if I go down two and to the left three, I'm gonna see that it passes through at this point. So if I connect those points with a line, it's gonna look something like this over here. Now, if you look at these two equations or these two graphs that we've ended up with, we've actually ended up with the exact same line. So basically what happens is that this equation over here two thirds X minus one and this equation that we wrote in point slope form are actually the same exact equation written in two different ways. And the way I'm going to show that to you is we're gonna use, we're gonna go through part C, we're gonna rewrite my equation in slope intercept form. All right. So how do I take this equation? And now write this in Y equals MX plus B I basically just have to solve and isolate for this Y term over here. So let's do that. I've got Y minus one equals two thirds X minus three. So if I want Y by itself first, I'm gonna have to distribute the two thirds into everything that's inside the par C. So I get Y minus one equals two thirds X and then two thirds times negative three into being negative two. All right. Now, the last thing I have to do is just add one to both sides over here, whatever you do to one side you have to do to the other. And so I just get Y equals two thirds X and then negative two po one becomes minus one. So notice how now this is in slope intercept form and we've basically just gotten right back to the equation that we had on the left. So again, these two graphs over here and these two equations mean the exact same thing, but they're just written in two slightly different ways. All right. So hopefully that made sense. Hopefully you understand the difference between slope intercept and point slope form. Let me know if you have any questions. Thanks for watching.

12

Problem

Problem

Write the point-slope form of the equation of a line with a slope of $-\frac25$ that passes through (1, 3). Then graph the equation.

A

$y-3=-\frac25\left(x-1\right)$

B

$y-3=x-1$

C

$y+3=\frac25\left(x+1\right)$

D

$y=-\frac25x-1$

13

Problem

Problem

Write the point-slope form of the equation of a line with a slope of $0$ that passes through $\left(2,-4\right)$ . Then graph the equation.

A

$y+4=x-2$

B

$y+4=x$

C

$y+4=0$

D

$y=0$

14

concept

Finding Equations of Lines Given Two Points

Video duration:

4m

Play a video:

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, that the line passed through X one Y one. What I'm gonna show you is that in some problems like this one, we're gonna work out down here. Sometimes you won't be given the, that end value and instead you'll be given information about two of the points. And what I'm gonna show you is that we're still going to use the point slope form for this. And it's actually gonna 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. All right. So I just want to remind you that we're not going to use the slope intercept form. Why it was MX plus 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 these kind in this type of problem. All we're told through is that it passes through these two points, negative one, comma negative five and two comma four. So we don't use Y equals MX plus B and instead we're still going to use the point slope form because we have some about the points. So I'm gonna write my equation Y minus Y one equals M parentheses X minus X one. Remember this equation, I need three numbers here. I need Y one, X one 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 wanna go, go find that first. So how do I go and calculate M or remember that the M is really just the slope, it's the rise over the run. Um So basically, you just use this, these, these two equations over here delta Y over delta X or Y two minus Y one over X two minus X one. So I'm gonna use this Y two minus Y one over X two minus X one because I'm told information about two of the points and I don't have these things sort of graphed out already. All right. So remember I want to pick my points. Uh so that the X one, Y one is the leftmost point. Again, it doesn't matter which point that you pick, it'll still be the same no matter what. Um So this is gonna be X one, Y one and this is gonna be my X two Y two. All right. So here, what I've got here to calculate the slope is I've got Y two minus Y one. So in this case, I've got four minus negative five. So I've got four minus negative five over here and then I've got divided by X two minus X one. So this is gonna be two minus negative 12 minus negative one. What I end up getting over here is I'm getting and getting nine divided by three. which actually just gives me an M of three. So that's my M term over here. I've got what my slope is and I'm just gonna pop that right back into this equation, but I'm not done yet because now I just need my X one and Y one. Now again, it doesn't matter which point that you pick as your X one, Y one. So you can use these numbers and pop them into this equation over here for X one, Y one or you could use these. It really just depends on which ones you picked now because I've already picked my X one Y one to be these numbers. I'm go ahead and use these numbers. OK? So what I'm gonna do here is I'm gonna do Y minus and I'm gonna use the points uh negative one comma negative five. So my Y one in this case, is gonna be negative five, right? That just comes from right over here. And then this is gonna be M which is three and I just calculated for the slope and this is going to be parentheses X minus X one. In this case, my X one was just equal to negative one. All right. So really, this is all there is to it. Now, I just have to tidy up this equation a little bit. So I've got Y plus five equals three parentheses X plus one. And if I go ahead and sort of just uh draw a little box around this, this actually ends up being my final equation. So this is the equation of this line um that passes through the points and, 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. Now, what we're gonna do is we're just gonna graph this. All right. Well, graphing this is actually pretty straightforward because remember we always need two points to graph a line and we actually already know what they are. This is just negative one, comma negative five and then two comma four over here. So all you have to do is just connect these lines or just connect these points with a straight line like this. If you have a straight edge, that's uh that's even better. But this is really all the 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. All right. So that's really the big idea here is that whenever you're given two points, you can just use either one of the two points that you're given as X one and Y one. It won't matter. All right. So that's it for this one. Folks. Let me know if you have any questions.

15

Problem

Problem

Write the point-slope form of the equation of a line that passes through the points $\left(2,1\right)$ and $\left(-4,3\right)$ . Then graph the equation.

A

$y-1=-\frac13\left(x-2\right)$

B

$y-3=-\frac13\left(x-2\right)$

C

$y=\frac13x-4$

D

$y-2=-\frac13\left(x-1\right)$

16

concept

Standard Form of Line Equations

Video duration:

5m

Play a video:

Everyone. So throughout a discussion on lines, we've seen equations and lines in different forms like slope intercept or point slope form. But you might be given a problem with an equation that kind of looks like this with all the terms of X and Y on the left side. But I'm gonna show you in today's video is that this is just another way of writing an equation of a line. This is called the standard form. I'm gonna show you what it's helpful for. But most of the time in problems you're gonna be able to take this equation and rewrite it. And in one of the forms that we already know like slope intercept, I show you that these two equations actually mean the exact same thing. They're just written in slightly different ways. All right, let's get started here. So if you take a look at our problem, we're gonna do is we're gonna find the slope and the Y intercept, which we already know how to do. But of this equation over here negative nine X plus three Y minus 12 0, it doesn't look anything like the forms that we already know. So the standard form. The way these, these equations sort of generally are written is sort of like A X plus by plus C equals zero. So clearly we can see here that this is sort of like A and B and C and this equals zero. So instead of having like MS and Bs and, and you know, and points and things like that, these are really just sort of coefficients that stand for the numbers inside of your equation. All right. So how do I solve this problem? I have A's B's and C's, but I'm asked for the slope in the intercepts. Remember that the slope is just M. So that's what I want to find. And I wanna find B over here. How do I find that from this equation over here? Well, basically what happens is that whenever you're given a problem in standard form, like we have here and you're asked for the slope or the intercept, you're gonna have to rewrite this equation. And the way that we rewrite the equation is basically just by isolating and solving for Y to the left side of the equation. All right. So I'm gonna take this equation over here and to convert it into Y equals MX plus B, I'm gonna have to rewrite it so that Y is on the left side. All right. So I'm gonna take this negative nine X and I have to move it over to the right side and have to add nine X and I have to take the negative 12 and also move it over and add 12. So I have to take everything and move it to the right side. What happens is the three Y just stays behind on the left side like a three Y equals and then both of these things become positive. So I have nine X plus 12. All right. Now, we're not done yet because this still doesn't look like Y equals MX plus B. I have to get rid of the three. And the way that I do that is by dividing it out. So I have to divide each number in the equation by three. So the three goes away on the left side and all I'm left with is Y and then we here, what happens with 9/3 is just, just turns it to three and then 12/3 just becomes four. So this is the equation now that I've ended up with Y equals three X plus four. And if you look at this, this actually is in slope intercept form, right? I've got Y equals MX plus B. So these are the two numbers, I've got three and four. And I'm basically done here. Those are the numbers I'm looking for my slope is equal to three and my Y intercept is equal to four. So that's the answer to this problem. These, this equation over here. And this one mean the exact same thing, they're just written in slightly different ways. All right. OK. So that's one of the cases where you use Standard Form. It's basically when you're just asked to rewrite it in a different form. But what I'm also gonna show you right now is it's also really helpful in finding the X and Y intercepts. So let's go ahead and take a look at our next example over here. So you might be given a line that's written in standard form like this three X plus two, Y minus six equals zero. And you might have to find the intercepts without first converting it back to slope intercept form. All right. So I'm gonna show you how to do that to graph a line in standard form. You can find these intercepts very quickly without rewriting it in slope intercept. And basically, let's just sort of recall what these intercepts actually mean. Remember that the intercept, the X intercept it where is where it crosses the X axis and it's where the Y value is equal to zero. And the opposite happens for the yer set. That's where the X value is zero. And right, so that's where it basically crosses the Y axis and that's where the X value is equal to zero. So what we're gonna do here in this problem is we're gonna take this equation over here and we're gonna set that Y and X equal to zero and solve. Let's go ahead and do this for the X intercept, right? So if I want the X intercept, what I'm gonna do is I'm gonna rewrite this equation. So three X plus two, Y minus six equals zero. And I'm gonna set Y equal to zero and then I'm just gonna solve for X. So in other words, I'm gonna take this three X and I'm just gonna replace Y with zero in which that whole term now just goes away. And now, all I have to do is I have an equation of one variable and I just solve for X. So how do I do that? Well, I'm gonna bring the six over to the other side. So this just becomes three, X equals six and X equals two. So that's my X intercept, right? I just solve for X. So what that means is if I go over my graph here, I can say that I know this graph crosses the X axis at X equals two. But that's not enough to graph the line because it's only one point. So now we have to do the exact same thing. But now for the Y intercept and once we have these two points, then we can go ahead and connect them and form our line. So with the Y intercept, I'm going to do the exact same thing. Three X equals two, Y minus six equals zero. But now what I'm gonna do is do the opposite. I'm gonna set X equal to zero over here. And I'm gonna solve for Y. So what I have to do is I have to replace the X with zero and that term just goes away. So this is uh two, Y minus six equals zero. And now I just move the six over like I did the other side. Um So that's plus six and I end up with two Y equals six. And if you figure this out, this is gonna be Y equals three. So these are my X and Y intercepts. I've got X equals two and Y equals three. So that's this point over here. Now, we have the two points of our line so we can actually just connect them and that, that's gonna form our line segment. So I just connect these two points using a ruler or a straight edge or something like that. And that's gonna be the equation of my line just gonna move it over a little bit. Um There we go. So this is the equation of my line and I got that without having to convert it to slope intercept form. All right. So hopefully, that made sense. Thanks for watching.

17

Problem

Problem

Find the slope & $y-intercept$ of the line given by the equation $3x+2y-6=0$

A

$m=2,b=-3$

B

$m=-\frac32,b=3$

C

$m=3,b=-\frac32$

D

$m=\frac23,b=2$

18

Problem

Problem

Graph the equation $9x+6y+18=0$ by finding the intercepts.

A

B

C

D

19

concept

Parallel & Perpendicular Lines

Video duration:

7m

Play a video:

Everyone. So in previous videos, we've seen the three different ways of writing line equations, three different forms. Well, some problems like the one we're gonna work out down below will give us an equation in any one of these forms and ask us to write lines that are parallel or perpendicular to those lines. That's what I want to show you in this video. We're going to see that parallel and perpendicular lines are really just related by the values of their slopes. All right. So let's go ahead and take a look here. So I've got these two graphs and I've got these two lines in this diagram over here, negative three X plus two and negative three X minus four. And you can see that they're graphed here. Well, if you notice that these lines are sort of like perfectly, almost identical, they're just sort of shifted uh you know, to the side. And that's the key characteristic of parallel lines. The key characteristic is that because they're sort of exactly the same and just shifted over, they actually never intersect. And that really just has to do with their slopes. Notice how the slopes here are both negative three. So for parallel lines, the slopes are always equal to each other and the Y intercepts are different. So in other words, they both have a slope of negative three, but their Y intercepts are different like two and negative four. If both of these things were the same, it would actually just be describing the exact same line. So the key characteristic for parallel lines is the slopes are equal and the BS of the Y intercepts are different. All right. So if these two things have the same slope, it's basically like the rise over the run of each of these points will always be the same and they'll never cross each other. All right. Now, let's take a look at perpendicular lines here. What I've got is I've got the negative three X plus two. I've got the exact same lines they had over here. But now the other line like orange one over here has a one third X minus four. So clearly these two things have very different slopes in Y intercepts. And we can see that they do actually intersect. Now, what's special about perpendicular lines is that the point where they intersect is they intersect at right angles. So in other words, this sort of this intersection point over here has a, an angle of 90 degrees. That's what perpendicular actually means. And basically what you need to know about these lines is that the slopes here are different. So the negative three is the slope of the one line. And then we have one third of the slope of the other. These things are related because notice how one is positive or sorry, one is negative and the other one is positive and they're also reciprocals of each other. That's actually always what's gonna happen between perpendicular lines, the slopes will always have opposite signs. So in other words, one was positive and one was negative and they're also gonna be reciprocals of each other. So basically just gonna flip the fractions or whole numbers or something like that. All right. So that's basically the difference between parallel and perpendicular lines. You may see some symbols regarding these things, you may see some like sort of double lines for parallel versus this other sort of like sign that kind of looks like an upside down t one way I kind of like to remember it is that parallel lines uh have the symbol and it kind of looks like the double L that's inside the word. All right. So just in case you see that that's basically the difference. Let's go ahead and take a look at some uh example problems. All right. So here, what we want, I want to do is we want to write an equation of a line that passes through a point and is parallel to some of the reference line that it's giving us Y equals two X minus six. So in other words, I want to write an equation of a line that is parallel to this line over here. So parallel means that I wanna have a, an equal slope and a different Y intercept. So in other words, I'm looking for a line in which the slope is equal to two because that's the slope of this line and equals two. So it has to be the same. All right. Now notice how this problem actually hasn't told us what form of an equation to write. And that's what we're gonna have to figure out. But notice how this problem tells us that we also are gonna be passing through a point. That's the equation of our line. So we're gonna be passing through the points negative one comma four. And if you remember what happens is if you're ever given the slope and a point, we always write the equation using point slope form. So we're gonna write the equation in point slope. All right. So I'm just gonna write that out for a second. Um Basically what we're gonna do here is we're gonna write Y minus Y one equals MRCX minus X one. All right. So this is the sort of now we just have to fill in the, the sort of variables and remember the Y one and X one just have to do with the point that we picked, which is negative one comma four and the M is just the two. So just going to fill this out really quick. I'm going to do the Y one which is four. So Y minus four equals M which is two parentheses X minus and this is going to be negative one. Remember these are variables and we don't replace them with numbers. All right. So all we have to do is just clean this up a little bit, but this is going to be Y minus four equals two parentheses X plus one. All right. So this is the equation of a line that passes through the 0.1 comma negative four or negative one comma four and is parallel to this equation over here because they have the same MS. And in fact, if you graph these two equations out, what you're gonna see is we're gonna have two X minus six, which kind of looks like uh this over here passes through negative six and has a slope of two. And then if you graph this point over here, we know that it's gonna pass through the point negative one comma four and it's also gonna have a slope of two. So if you draw some points out and connect them, we're gonna see sort sort of as a rough sketch that these two lines are parallel to each other, they will never intersect. All right. Let's take a look at our next problem here. Here, we have to write the equation of a line that is now perpendicular. So we're looking for a line that is perpendicular to this equation over here that's written in standard form and it has a Y intercept of three. So what are we told we're told that it has a Y intercept which remember of B, so B is equal to three. All right. So remember which. So first we have to figure out what form we're gonna write this equation in. And remember that whenever we're given a B term or asked for B, we're always gonna write this in slope intercept form, right? So for given or asked for B, we write it in slope intercept form. OK. So if I want to write slope intercept form, I need the B which I have, but I also need the M so that M, that I need has to be perpendicular to this line over here. In fact, what I'm actually gonna do is I'm gonna write this over here for seconds. So I have to have am that is perpendicular to this line which is in standard form. So to figure that out, first, I'm gonna have to sort of isolate and solve for this Y over here. What I'm gonna do is I'm gonna take this X, move it over and the negative eight and move it over. I'm gonna have to subtract X and I'm gonna have to add eight from both sides, all right. So this ends up going away and I end up with four Y equals negative X plus eight. Now I have to do is I have to divide by four of each one through each one of the terms over here. And I get rid of the four. So this is gonna be Y equals negative 1/4 X plus and then 8/4 just becomes two. So remember this is the equation, not that I'm trying to find this is the equation of this line over here just written in standard form. And I needed to write in standard form because I needed to figure out what the M is. All right. So this M is negative 1/4 to get an M my line that I'm trying to find, I have to do the opposite sign. And the reciprocal of this number, the opposite sign is gonna be a positive and the reciprocal of this number is going to be an M that's equal to four. So now that I have M and B now I can go ahead and write the equation of my line that is perpendicular. So this is gonna be ye equals MX plus B. So in other words, this is gonna be Y equals uh four X and then plus three over here, let's graph these lines and see if we've actually done this correctly. So this is an equation Y equals negative 1/4 X plus two. So it starts here and we go down one and over four, up one and to the left four. So this is, this line is gonna kind of look like this. Now, our line is gonna look like four X plus three. So that's an equation that starts here and it has a slope of four. So we go up 4/1 or down four and then over one, down 4/1. So this line is gonna look kind of like this. And if you look at these two lines over here, they intersect at this point and the intersection is definitely 90 degrees. All right. So that's it for this one, folks. Hopefully, this made sense. Thanks for watching.

20

Problem

Problem

Write an equation of a line that passes through the point $\left(3,-4\right)$ and is parallel to the line $x+2y+18=0$.