Algebra 1 Chapter 6 Lesson 1
Algebra 1
In lesson 6 one we're solving systems of equations by graphing. Now, a system of equation is exactly what it sounds like, a system has at least. Two equations. So it just made up of two equations in one single prom at least. You could have three, four, 5, 6, but we'll get to that later. Now, the answer is where the equations. Intersect. Now, what do I mean when I say intersect? Well, if I just quickly sketch here, if we have a line going there in a line coming here, the point of intersection is going to be let me zoom in on this. The point of intersection is going to be right there on that red dot, okay? So it's going to be where they were those two cross. Now, because it is where the equations intersect are answer. There are solution. Is going to be a point, so we need to make sure we give our answer as X comma Y or a point. You can already maybe guess what if we have let me erase this instead of a graph where we have two lines that intersect clearly what if I have that line there in this line here parallel lines, well if our solution is where our equations intersect, do parallel lines ever intersect? No, they never do. So in that case, we wouldn't have any solutions. Now, what if we had. A line here and a second line right on top of it? Now I want you obviously I'm not around to have it on this point, but assume that they are right on top of each other. Well, how many points are those lines touching? That's what I mean when I'm saying intersect is where do they touch? Well, if they're right on top of each other, that means they're touching at every single point that makes up a line. The line is made up of an infinite number of points. So we would have an infinite number of solutions there. Now, we have here a table that we're going to talk about the number of solutions. So we've talked about a case where we've had one solution. We've talked about a case where we've had infinite. Solutions. And we've talked about a case where we've had no solutions. Now each one of these has a specific name to it. The name for one we have one solution is consistent. In dependent. The name for when we have an infinite number of solutions is consistent. Dependent. And when we have no solutions is going to be in consistent, now I want to talk about some easy ways to remember these names because yes, you are going to have to name these. Pretty regularly on a regular basis, you're going to have to name these things. When we have one solution, the one way that you can remember this first part consistent, we'll notice that when I have one solution or infinite solutions, I have the word consistent in there. Whenever you have any system that has at least one solution, and I say at least with emphasis for reason, here I have one solution here I have infinite infinite clearly more than one. So whenever I have at least one solution, whenever there is a solution at all, it's going to be consistent. Obviously over here I have no solution. So instead of consistent, it's inconsistent. So if I have a solution at all, it has to be consistent. Now, here's where we tell the part between one solution and infinite solutions. If I have one solution, it's one answer. It stands by itself. And that answer is independent. If you're an independent person, you don't depend on anybody else. You stand by yourself. So one solution, yes, there is a solution, therefore it's consistent, and there's only one, so it's independent. You might want to put your own little extra notes in there. So go ahead and hit the pause button if you have to. Our infinite solutions, yes, we have a solution, so it's consistent, but now we have two lines that are dependent upon each other. They depend on each other. So it's dependent. So we have a solution, so it's consistent. And they're dependent on each other. And we already talked about no solutions. If consistent means we do have solutions. And here we have no solutions. That means it's inconsistent. Now let's take a look at what the graphs look like. And this is, again, we already drew this kind of sketch these. If we have one solution, we're going to have a line coming like that. And it could come anywhere, but they're going to kind of be an X where your answer is going to be where they intersect there in the red. We have one solution. It's consistent and independent. For infinite solutions, you're going to have a line coming here and you're going to have a second line that's going to be right on top of that. Hopefully you can draw that a little better so it looks like it's right on top of it. Let me see if I can do a little better here. I think that's going to be as close as I can get. So an infinite number of solutions, it's intersecting constantly. It's consistent in the two lines because they're right on top of each other are dependent upon each other. And our last case is going to be, like I said, when we have two parallel lines because parallel lines will never cross. It's going to be inconsistent. We have no solutions. Let's take a look at a couple examples here. We're going to use the graph in the graph that we're going to use is right here to the right. Use a graph to determine the type of system also report how many solutions and if we have one solution name it. That's a lot of directions. So we're just going to take it one step at a time. Let's look at part a here. Y equals negative two X -5 and Y equals negative two X plus three. So let's look at our graph here. So we have Y equals negative two X -5, so that's this red one here. And Y equals negative two X plus three. That's this green one. So I realize that there's this blue line and purple line, but I want to ignore those. Let's look at just this red line, just this green line. What can you tell me about these lines? Well, you can tell me that they're not crossing at all. But that doesn't necessarily mean that they are parallel. Take a look at this blue line and this purple line. They aren't crossing anywhere, however, as we can see, these lines go on forever. These lines will clearly cross somewhere down in this region if we were to continue this graph, okay? So the blue line and purple line clearly are not parallel. So we need to remember, and this is where we, again, we're building upon our previous lessons. What makes a parallel line? Parallel lines are made up of the same slope. I'm looking at the red line in the green line. I'm looking at their equations, which I had their equations over here. They're slopes are right in front of the X because they're both in Y equals MX plus B form. The slopes are exactly the same. They're both negative two, so that's how I know. That, this system is in consistent. And it has no solutions. So again, I'm looking at my directions here. I'm using the graph to determine the type of system I set inconsistent. And I need to report how many solutions has no solutions. The last bit of direction says, if only one solution name it, well, we just said that we have no solution. So we don't need to worry about naming it because there are none. Let's look at part B here. Y equals two X plus three, so I'm looking at my graph, so that's going to be this blue line like was two X plus three and Y equals negative two X plus three, which is going to be this green line. So let's zoom in here. We're looking at the blue and the green. And we're trying to determine where they cross. Well, it looks like they cross right here. So they only cross one time, so looking at our chart, we know that whenever we cross one time, we are consistent. Independent. Because we have at least one answer, there is an answer. So therefore, we're consistent. And we only have one, and the number one tells us it stands by itself. It's independent. So we now have named it. We've determined the type now we need to report how many solutions. So we know consistent independent means there is one solution. And since there is one solution we need to go ahead and name the solution when we name it by just saying, hey, where do these bad boys cross? So it looks like they don't go over any. So let me get rid of that and give my pointer. We don't move side to side at all because it's right up here. So it's going to be zero comma one, two, three. So zero comma three is going to be our solution. So that would be our answer for part B up here would be our answer for part a now, what happens when we have to graph these? Well, that's no big deal. We're going to solve this system by graphing, then we need to name the system and the number of solutions, and if only one solution name it so it's the exact same directions as the example one. However, this time, instead of looking at drawings, you need to make the drawing. So let's get a graph here. So there's a graph that we can use that. I need to look at these two equations I have X minus Y is equal to two and two X plus three Y is equal to 9. Now, the important thing to realize here is we don't know how to graph out of standard form, which both of these are in. We know how to graph out a Y equals MX plus B form. So we need to get these two equations in that form. Well, the only way to do that is by looking at these two equations separately. So let me I'll leave that there. Let me just bring it down. Love and second copy over here. So that way we can mess with it and whatnot. I need to do this Y equals MX plus B so I need to move my X so this is a positive X, I'm going to subtract X from both sides. It's going to go away. Negative Y is equal to negative X plus two. This Y is not by itself. I need to divide everything by negative one. So those negatives go away. Now I have Y is equal to negative X divided by negative one is going to be positive X, positive two divided by negative one is going to be negative two. So there's Y equals MX plus B form for that first equation. Now let's do the same thing with the second equation. And we'll put this one in this kind of bluish greenish color. This is a positive two X, so I'm going to subtract two X from both sides. So I have three Y is equal to negative two X plus 9. Divide by three divided by three divided by three. I have Y is equal to negative two thirds X plus three, 9 divided by three is three. So I need to graph these two. I already have my blue pen up, so let's go ahead and use that. So I'm going to go up three first. So let's zoom in here a little bit. One, two, three. And my slope was negative two over three. So down to over three to the right. Down to over three to the right. Down to over three. And no matter. So I'm going to go the other way up to over three to the left. Up to over one, two, three, up two over one, two, three. And I can draw my line there. Now let's bring up the purple equation. So we have Y equals X minus two. Let me get my purple panel. We'll zoom back in here so we can try to draw as accurate as we can. So as X minus two, so I'm going to go down to the start. So here's where I'm starting. And now it's just X so my coefficient is one over one, so I'm just going to go up one over one in every direction. So there, there. There, there. And we'll go the other direction as well. And we'll connect those dots. Okay. So there we have our graph. Now, we need to name this type of system. We name it by asking ourselves how many times does this cross? Well, it only crosses one time right here. So when we come to name this, we're going to go ahead and name this thing as consistent. Independent. We're going to also tell the number of solutions, so consistent independent, one solution. And it also says, if only one solution we need to go ahead and name it, so that one solution looks to be. At one, two, three, comma one. So now we just need to three comma one. So that would be our answer. As well as obviously we need it. We would need to have our graph shown as well to show our work.