Common Core Geometry Unit 5 Lesson 3 Equations of Lines

Hello and welcome to another common core geometry lesson by E math instruction. My name is Kirk weiler, and today we're going to be doing unit number 5 lesson number four on the equation equations of lines. All right, now almost everything that we do in today's lesson is going to be review from common core algebra one. And that was probably kind of review from 8th grade math. So there's going to be a lot of review in today's lesson. On how to look at the equation of a line, how to interpret it, how to plot it, what the intersection points of equations of lines mean. And we'll also try to bring back things like parallel and perpendicularity from the last couple of lessons. Let's jump right into it and get into a little bit of the review of this subject matter. All right. Now you've been seeing equations of lines for quite some time. Most of you have actually been probably seeing them since about 5th grade. Granted, you may have only been plotting on sort of this portion of the axes back then, but you did see these kind of relationships. So let's jump right into exercise one. Without the use of your calculator without the use of your calculator, identify the slope and Y intercept of each of the following lines and then plot them on the grid shown. Label each with its equation. Well, even though we haven't done any line plotting per se based on equations, this is not a bad point to ask you to pause the video and see what you remember about equations of lines and how it can help you plotting them. So take a few minutes now and see what you can remember and then obviously we're going to come back to it in a bit. All right, let's go through them. Whenever we have a line that's written in what's called Y equals MX plus B form. The coefficient on the X is the slope. So in this case, the slope of this first line is two. And if you want to think about it in terms of rise and run, right? You'd maybe want to plot it like this. Arise of two and a run of one. Okay. It's Y intercept though. Where it intersects the Y axis. Is the constant that's added on. All right? So before we plot the second line, let's go ahead and take a look at this one. It's pretty easy. What we'll do is we'll come in and we'll plot the Y intercept down here at negative 7. And then we'll use the fact that the slope is two over one to do a run of one and a rise of two and a run of one and a rise of two. This picture is going to get kind of ugly soon if I keep doing that. Now as we've talked about extensively in this course, of course, once you have two points on a line, technically that's all you need. But we know how inaccurate line plotting can be and things like that. So it's kind of nice to have a variety of them. Before we take our straight edge out and let it let it do its thing. So let me now kind of connect these things. All right, like that. And get rid of this for now. Double arrows, okay? And I want to label it with its equation. It is very, very important when you're plotting more than two things on the coordinate grid that have equations that you label them with their equations so that somebody who's looking at your work can properly assess which one is which. Let's take a look at the second equation now, wherever it is there it is, and see what we can say about that. In this case, the slope is negative one half. Now I can interpret that then, right? Again, rise and run. And from a previous lesson, you know that probably what this means is that these two lines are perpendicular to each other. That's not really part of the problem. The Y intercept is three. So again, just like before, let's go in, Y intercept of three. Let's go with a run of two and a drop of one, a run of two, and a drop of one. And again, I think I'll stop doing the arrows right now. We can always kind of reverse direction. You certainly don't have to plot all those points. You know, it kind of really is all about how many you need to feel comfortable with drawing a good, good, straight line. All right? I would always suggest more than two just so that you can be guaranteed to have a good graph. Let me label that one with its equation Y equals negative one half X plus three. All right. Great. There are my two lines. The coefficient on the X, the number multiplying X is the slope. And the constant added on is the Y intercept. Remember, the slope is not two X it's just the two. Slope is a numerical measure of rise divided by run. It doesn't include the X it's the coefficient on the X let's take a look at the second part of this problem. This is also from algebra one. It says, what is the solution to this equation? And why? Right? Now, I'd like you to think about this a little bit. But I don't want you to do the algebra to solve the equation. It'd be a little messy. I'm sure you could do it. Maybe a little messy because of the fraction negative one half. But you should be able to solve this equation somehow by using this graph. Take a moment and think about how. All right, well, let's talk about it. Notice that the two X -7 let me get rid of that larger circle. Is the Y equals equation that we graphed to begin with. And this is the second one that we graphed. And ultimately, when we're solving this equation, what we really are saying is find the X coordinate that makes these two Y coordinates equal to each other. And that will always happen, in fact, by definition that will happen. At the intersection point. Now I'm not asking for the intersection point. This equation only has X in it, and I only want to solve it for X, but the X coordinate where these two lines intersect, right? The coordinate point itself is at four comma one. But the actual solution, then, is X equals four. And it is the X coordinate. Where. The two lines intersect. All right. Just a little reminder that you can solve equations graphically by graphing one side graphing the other side and then finding the X coordinate where they intersect or X coordinates if there's more than one. But with two lines, they can at best intersect once. All right, let's keep going. Okay. Now, there are simple lines, and there are more complicated lines. What I wanted to do in exercise two is look at two extremely important lines. All right? That are also extremely simple. And at the same time, try to make a point about lines. No pun intended. All right, let me switch back to blue real quick. Okay. And let's talk about the lines. Y equals X and Y equals negative X both of these lines are going to be very, very important in this course, and they're both very, very simple lines. But let's see if you can answer a letter a letter I asks, what are the slopes of these lines? All right, so M equals and M equals. Why don't you go ahead and see if you can figure that out. All right. Well, the slope of Y equals X if you really, really absolutely need to have it in MX plus B form. You could write it as Y equals one X plus B in which case the slope is one, or you could say it's one over one. All right? Likewise, this line, if you really had to, you could write it as Y equals negative one X plus zero. I should have had zero here. All right. In which case the slope is negative one or negative one divided by one. Watch out. It's definitely not negative one divided by negative one, negative divided by negative is a positive. So that would be incorrect. All right. Now, letter B asks us to write some, maybe I should put it in quotes. Obvious coordinate points that lie on both lines. Now, this is important. A lot of students learn how to plot lines using the Y equals MX plus B approach, and they say, okay, I'm going to plot B then I'm going to take in and I'm going to go up to and over three and up to and over three, et cetera. And there's nothing particularly wrong about that. But with simple lines, like Y equals X, Y equals negative X, right? We also want to be able to interpret what the equation is saying. You see, this line, Y equals X is essentially saying plot all points where the Y coordinate is equal to the X coordinate. So for instance, the .1 comma one lies on that line. The point two comma two lies on that line. The point, negative one, comma negative one lies on that line. Okay? Why don't you flesh out a few more and also do some for Y equals negative X. All right. Well, again, we could do a bunch of them here. Zero comma zero, negative two, comma negative two, et cetera. All right? This one, maybe a little bit harder to interpret, but remember what that really is saying is that the Y coordinate should be the opposite of the X coordinate. Again, opposite is kind of a loaded word. But for instance, if the X coordinate was, let's say, one, the Y coordinate would have to be negative one. If the X coordinate was two, the Y coordinate would have to be negative two. Let's go in reverse. How about if the negative three positive three how about negative four positive four? Lots of different points, right? As well, zero comma zero. Think about that for a moment, right? What is negative zero? Well, that's still zero, because negative one times zero is still zero. So now, letter C create a graph of the lines in the grid provided label with their equations. All right. Even if we didn't say label with their equations, we would anyway because we have two different graphs here. So here we go. Let's do Y equals X together. Again, we can just plot a bunch of points. We know zero, zero lies on it, one, one, two, two, three, three, negative one, negative one, negative two, negative two, et cetera. Right? And then four. Still not quite there. There we go. Don't you hate it when your ruler is not quite long enough. All right, there it is. Y equals X very, very important line. If you need to plot Y equals X, you should be able to plot it just immediately. Right. Zero zero one one two two three three. The same for Y equals negative X, right? We still have the zero zero point, but then we have negative one, one, negative two, two, negative three, three. Et cetera. And we get a line that looks like this. These two lines are going to become very, very important for us when we do some work with reflections across lines. And you want to be able to plot them very quickly and accurately. That any issues. All right, and we get white balls. Negative X all right. They're kind of classic lines. This one, if you will, kind of, bisects that 90° angle. And I guess this 90° angle, and Y equals negative X, that bisects this 90. And this 90. All right, two exceptionally important lines. I wanted to take just a moment in this lesson to kind of plot them for the first time so that we have them down. Take a moment though. Let me move mister ruler here. And write down anything you need to. All right, let's move on. Great. Now, for whatever reason, many times. You're presented with equations of lines that are not in Y equals MX plus B form. And it is very important to be able to do the relatively simple algebra. And it should be at most two steps of algebra. To get it into Y equals MX plus B form so that you can identify the slope. If you need to and the Y intercept if you need to. So let's go through a little bit of that algebraic manipulation in this next problem. Exercise number three. For each of the following equations, rearrange it into Y equals MX plus B form and identify the slope and the Y intercept of the line. Okay. So in none of these three examples is the thing rearranged to look like this. All right, but we've got to get good at doing this because often times for whatever reason, lines are given in this form instead of in this form. Let's do letter a together. Maybe even do letter B together and then have your work on C on your own. Let array is very, very easy, because when we look at it, the only thing that's on the left hand side of the equation hanging out with the Y is this positive two X and we can easily move it to the other side of the equation. I'm just going to write this down again. By simply subtracting two X from both sides. Now, that would normally leave you with something that looks like this. And there's nothing particularly wrong with this, but students can get confused. Some students will think, well, the first number I see must be the slope and the second number I see must be the Y intercept. But remember, it's really all about the coefficient of X being the slope and the constant being the Y intercept. So many people could probably take a look at this and say, oh, okay, this and this. If not, you should use the commutative property of addition. To swing the surround and write it like this. In fact, every time I will. Because once I have it looking like this, I can then easily say granted, I don't have a lot of room left, but I can easily say that the slope is negative two. And the Y intercept is 7. Okay? Let's do another one. So sometimes that's it, though. That's all you have to do to get the Y by itself. On the other hand, if we look at letter B, what we see is we've got two X plus three Y equals 12. Now, we've got both a two X hanging out with the Y and we have it multiplied by three. Now one thing that you could do right away if you wanted to is you can use the commutative property of addition. To rewrite the left hand side like that. You don't have to. But you can. This then allows me to easily subtract two X from both sides. And right away, I'm going to write this as negative two X plus 12. Remember, that's a positive 12. If it had been a negative 12 sitting there, then it would be negative two X -12. All right? Now we're getting closer, but we need to have one Y not three Y so now what we'll do is we'll divide both sides of this equation by three. And remember, just like multiplication, division distributes. We don't get to just divide the nice things by three. We have to divide everything by three. Let me go with a couple of steps here. That means we would have negative two X divided by three. And we'd have positive 12 divided by three. Ultimately speaking, right? Then I would have Y equals negative two thirds X and of course 12 divided by three is positive four. So that means that my slope will be negative two thirds. And my Y intercept will be positive four. And now I have my answer. Okay. Important. To be able to get everything on one side of the equation, have it just be Y equals so that you can properly identify the slope and the Y intercept. We got one more left. I'd like for you to work on letter C on your own. Take a few minutes. Are you ready to go through it? Let's do it. Now again, similar to letter B, we've got four X -6 Y equals 7. And we've got quite a bit that we have to do. Again, you don't have to do this, but I think it's helpful to rewrite the left hand side immediately. As negative 6 Y plus I've got a little bit of a lag on the board right now, which is interesting. Plus four X is equal to positive 7. All right? What I'm going to do is I'm going to get rid of the four X by subtracting it from both sides. All right. I'll get negative 6 Y is equal to negative four X plus 7. Now I have to get rid of that negative 6. So I'm going to divide both sides by negative 6. This is going to get ugly. And I'm going to get Y equals, that's good at least. Negative four X divided by negative 6. Plus 7. Divided by negative 6. And this is, this is ugly. We got fractions everywhere, negatives everywhere. It's not pretty. But at the end of the day, it is what it is. And here's what I mean by that. Let's kind of finish this up. We'll have Y equals negative whoops. Let me get rid of that. I'll have Y equals negative four over negative 6 X and then I think what I'll do is I'll just make that negative 7 6th. I'm going to change that negative divided by a negative into a positive. I'm also going to reduce the four 6th into two thirds. So although it took a lot of work, we got all the way there, which tells me that the slope, I apparently really love two thirds. The slope is two thirds. And the Y intercept is the negative 7 6th. Okay. It'd be lovely if every time we had to plot a line or identify its slope or its Y intercept. It was immediately written in Y equals form. Okay? But oftentimes, it's not the case. And we have to have confidence that when we look at an equation of a line. And we know it's the equation of a line because neither the X nor the Y has been squared or raised to any higher power, we have to have confidence so that we can rearrange those equations. And get them into Y equals form. All right. Take a moment and write down anything you need to here. Okay, let's move on. Now scrubbing the text this year. It just stays there. All right, now let's kind of get into a little bit more work with lines. Let's take a look at what extra size four asks. On the grid below, the line S is graphed, as well as the .2 comma four. Letter a, draw a line parallel to S that passes through two four, write the equation of the line below. All right, this is cool. This puts together a lot of what we've been doing both in this lesson and a couple of lessons ago when we looked at parallelism, all right? I want to draw a line that's parallel to S and passes through that point. Okay? So take a few minutes and see if you can do that. All right, well, first things first. A little review. Two lines are going to be parallel if they have equal slopes. So I got to figure out what the slope of S is. And I think I'll just do that graphically. Let me take a look at us. And what I see is if I pick a nice point like here, and I just kind of keep looking at it. What I can see is that for every two units I run, I fall one unit. So S. Has a slope of negative one half. Which means the line that's going to pass through this point also has to have a slope of negative one half. And let's give it a slope of negative one half. Let's come over here and let's run two units. And drop one unit. And then run two and drop one, et cetera. All right, that's probably plenty. Bring my ruler up. Rotate it down. That's pretty good. Okay. Now the question is, what's the equation of that line? Well, remember, we want the equation to look like this. Y equals MX plus B hey, it's been a little while since my random red came up. I just erased that and put it back and below. It's still red, but over here it's blue. We're just going to let the red be hang out. I don't even know why it's there. Anyway, so we needed Y equals MX plus B okay. We already knew what the slope was. It's negative one half. And of course, the B is the Y intercept. And what is that Y intercept? Well, it's 5. We can just look at it right on the graph, okay? So the equation is very simple. Y equals negative one half X plus 5. That's it. All right, let's deal with some perpendicularity. Letter B, draw a line, perpendicular to S that passes through the point to comma four. Write the equation of the line below. Why don't you take a few minutes and go ahead and do this. Remember, you'll have to go back to what we learned about perpendicular lines and how their slopes relate to one another. Good. All right, let's talk about it. Well, remember, the slope is a perpendicular lines are negative reciprocals of one another. In other words, you take your slope, you flip it, and you change its sign. So in this case, that slope of the perpendicular line will be a positive two over one, we're just two. Now that's going to be enough to allow me to plot that line. Maybe I'll go in red. Which then will probably mean that magical blue color will just show up somewhere. But let's go ahead and do it. Let's come over to our point, which, you know, I kind of lost a little bit. And let's do a slope of two, so that means I go over one. And up to, and then to the right one, and then up to, right? And I get, hopefully, more points. Like this. All right, that's gonna be plenty. Let's bring my ruler back up. Bring it down. Rotate. I don't know why I have to walk you through all of the little different things I have to do to get my ruler to work. Hopefully you just have a nice normal non virtual ruler. All right. Go back down like this. Oh yeah, yeah. I don't know what just happened there. Go back to blue. Lost my little two there. Okay, I kinda looks like a two. All right, well, let's take a look. We need to write its equation. What do we need? We need it slope, which we know, this too. And we need its Y intercept. Well, it's Y intercept is actually the origin. It's zero. So you could write the equation of this line in two very different well, not very different ways, but slightly different ways. You could either say it's Y equals two X plus zero, which many students will prefer because they want that Y equals MX plus B form, or what I would certainly prefer as a teacher, is just to see students writing down Y equals two X in other words, every point on the red line is one where the Y coordinate is simply twice the X coordinate. Even the zero zero, if you think about it, although it seems a little bit weird, but you can even see it, right? Here we've got the .2 comma one, here we've got the .4 sorry, here we've got the .1 comma two. Here we've got the .2 comma four. All right. And every case the Y coordinate is always just twice the X coordinate. But that's it. And again, the nice thing about this problem is it allows us to review the parallelism and perpendicularity conditions. Plus, get that Y equals MX plus B into the mix. All right, let me step back. Let me toss some ruler out of there. Maybe kind of bring this up, so maybe bring this up. So you can see the work. Step out of the way and copy down anything you need to. All right, let's move on. One more problem. All right, kind of a simple wrap up problem. Okay? Exercise number 5, which of the following lines is perpendicular to ten X plus 6 Y equals 12. Okay? One of these four lines is perpendicular to that one. Which one is it? Take a little bit of time. All right. Well, what we know is that when we have two lines that are perpendicular, we get those slopes that are negative reciprocals, negative reciprocals. But that means I've got to be identify the slope of that line. And that line is not in Y equals MX plus B form yet. So I got to work a little bit. Let me take this thing, bring it down here. Maybe just rewrite it ten X plus 6 Y is equal to 12. All right, maybe use the commutative property of addition to rewrite it as 6 Y plus ten X is equal to 12. Subtract off ten X from both sides. Watch out. Don't combine those two. This one doesn't have an X, this one does. So that really I'm just moving something to the other side. Y equals negative ten X plus 12. Divide both sides by 6. Gives me Y equals negative ten 6th X plus two. Think about that for a minute. I just did 12 divided by 6 and negative ten divided by 6. And now let's reduce this one. By dividing both numerator and denominator by two. Now again, be careful. It would be easy enough to look at that particular equation, go up to here and go, there it is, choice one, negative 5 thirds X plus two. I just got it. I just got it. Except that that wasn't what I was looking for. I'm looking for a line perpendicular to this one. That's the exact same line. It's not perpendicular to this one. It's not even parallel to this one. It's the same line. Just in a different form. My purpose was to really come in, look at this and say, ah, the slope of my line is negative 5 thirds. So therefore, a line perpendicular to it will have a slope of positive three fifths, and that would be choice four. Now what some students will get confused by is let's say, yeah, but what about the two? That's a two over here, and it's a negative 7 over there. Perpendicularity has nothing to do with the Y intercept. It's all about the slopes. It has nothing to do with where it intersects the Y axis. So yeah, this line intersects the Y axis at a Y coordinate of two. This one intersects the Y axis at a coordinate of the negative 7. That's completely okay. What matters is that relationship between their slopes. Which we could only identify. If we rearrange this into Y equals MX plus B four. All right. And again, out of the way, again, take a little bit of time if you need to. Okay, let's move on. Move on and wrap up. So today, we just really reviewed, right? We looked at the classic Y equals MX plus B form of a line. That's called the slope intercept form of a line, because when written in that form, you can easily identify the slope of the line, rise over run, and the Y intercept of that line. Where the line crosses the Y axis. All right? Simply by knowing that, and also knowing skills like how to rearrange lines to get them into that form, we can both plot lines and we can look at the perpendicularity and the parallelism of lines. That piece is a little bit new to this course. Everything else you saw back in algebra one. All right, we're going to work a lot more with this in the next lesson. For now let me just thank you for joining me for another common core geometry lesson by E map instruction. My name is Kirk weiler, and until next time, keep thinking. And keep solving problems.