 We say a system of equations is consistent if it has a solution. Now, when we say it has a solution, we mean there has at least one solution. It's more than zero, right? There could be one solution, two solutions, three solutions, 12 solutions, 5,280 solutions, infinitely made solutions. It has at least a solution. We call that a constants system. Now, if a system's not consistent, then it's inconsistent, which would mean it has no solution to the system whatsoever. Now, you'll often see when someone describes no solution, you'll see them draw a circle with a line through it. This is actually a notation that represents the empty set. That is the set that doesn't contain any elements. And in this situation, we're describing the solution set to the equations here. The solution set contains nothing, hence it's an empty solution. Now, if a linear system is consistent and it has only one solution, we refer to that as the independent case. Now, if a consistent linear system has multiple solutions, we call that the dependent case. Now, the terminology between independent and dependent, what does it even mean here? At the moment, we're not really gonna define what that means exactly. I'm trying to piggyback off of vocabulary that students probably picked up in a previous algebra class, like elementary algebra, intermediate algebra, advanced algebra, high school algebra, college algebra, whatever your prerequisite to linear algebra was. You probably talked about systems of linear equations. You solved them using substitution or elimination. You could solve them graphically as well. Feel free to use those techniques as we try to solve systems of equations in this lecture series, although we will use better solution techniques in the not too distant future. In that situation, the unique solution was described as this independent case. And the dependent case was where you had multiple solutions. When it comes to systems of linear equations, there is a trichotomy on the solution set, only three possibilities. There is, of course, possible to be inconsistent, and I'll show you some concrete examples of such a thing. You could have the inconsistent case which represents there is no solution whatsoever to the system of equations, no solution. The consistent case, on the other hand, breaks out into two possibilities, in fact. So the first possibility is that your solution set contains exactly one element. And we actually saw that type of example in the previous video. And this is what we mean by this independent case. Who are we independent from? Did the Americans declare independence from England? Well, I mean, although they did, that's not what we're referring to right here. This is actually describing a property known as linear independence, which will make more clear what that means in the future. For a linear system, the only other possibility is that you're consistent and you have infinitely many solutions. Because what one can show is that if a linear system has two solutions, you can actually algebraically generate more and more and more solutions without bounds. So you get infinitely many solutions. And this is what we mean by the dependent case. And so what I wanna do next is using two-dimensional examples. I wanna illustrate these three possibilities. So if you take a two-by-two system, it has two equations and two variables. Let's call them x and y. So we can think of these as points in the Cartesian plane. The first equation is gonna be 4x plus 2y equals 32. If we put that into slope intercept form, maybe to help us graph it a little better, you'll get y equals negative five over 2x plus 16. So we can see here the yellow line, the y intercept is 16. And then every time you go down five, you go over two, that's our yellow line. Our green line, if we put it in slope intercept form, we'll end up with y equals negative one half x plus eight. So we see a y intercept of eight. And every time we go down one, we go over two, something like that. And so the green line is not as steep as the yellow line, but still these two lines will intersect at a unique point in which this case, that point will be four, six. I'd encourage the viewer to verify that four, six is in fact the algebraic solution to the system of equations. But it's also the geometric solution. It's the only point that lies on both the yellow and the green line. It's the intersection of the two lines. And so since we have one solution, and there is a solution so it's consistent, it's a consistent linear system. And this is what we mean by this independent case that we were talking about in the previous slide. As another example, take the linear system y equals negative x plus five and two x plus two y equals three here. Now you'll notice that the first equation is actually already in slope intercept form. A linear system doesn't have to have all of the linear equations or any of them in fact, in the normal form where all the variables on the left-hand side and all the constants on the right-hand side. It turns out that when we work with linear systems, that's for the most part is how we would prefer them to look. But when it comes to grafting lines, we actually might prefer the slope intercept form in this situation, because we can see very quickly that the y-intercept that this yellow line would be five and the slope would equal negative one. So we go down one over one to draw the line here. The other equation which is not in slope intercept form can be put there very easily. Subtract two x from both sides and divide by two. You get y equals negative x plus three halves or one and a half. That gives us the y-intercept and the slope's the same. You're gonna go down one over one to get another point on the line. But actually kind of carrying off of that last idea a little bit more, looking at the slopes of these lines, we get negative one, negative one. The slope of these two lines is negative one. And since the slopes of the two lines are equal to each other, we would say that these two lines are parallel to each other. And geometrically speaking, if two lines are parallel, that happens if and only if they never intersect each other. So there's no point, that's on both the green line and the yellow line simultaneously. And so if there's no intersection between the lines, geometrically speaking, that means that the system of linear equations has no algebraic solution. So we see that there's no solution to this system of equations. And therefore we can conclude that this is an inconsistent, an inconsistent system of equations. Be aware that if you're trying to solve a system of equations, and it turns out there's no solution, the answer then would be, oh, I can't solve the system, not because I'm incompetent, but because there is no solution, you would report it's inconsistent with the evidence on why you decided it's inconsistent. Like in this case, we see these two parallel non- And the third possibility for the two by two case is a little bit fishy, but we'll investigate it nonetheless. Take the equation x equals 2 thirds, y plus three. This isn't in slope intercept form as we're used to. We actually could graph this the way it is, because we can see the y- the x-intercept here is gonna be three, and this is gonna be run over rise. Now, if that makes you feel uncomfortable whatsoever, we can deviate away from that. But what this tells us is that the y- the x-intercept of this line will be three. And then every time we go over three units, we go up two. But you can also put this into slope intercept form. We're gonna subtract three from both sides, and this would give us y equals three halves x minus nine halves, like so. And then the second equation, three x minus two y equals nine there. Notice that you'll subtract three x from both sides and divide by negative two. And so we get y equals, in that case, three halves x minus nine halves. Wait a second, these two equations are one and the same thing, right? These equations are equal to each other. And so as you try to graph the first line, you're like, oh, here's my line. And then when you do the other one, it's like, here's my line again. And if you weren't paying attention, it's like, where's the second line? And that's because these two lines are overlapping each other entirely. And so any point which is on the first line will automatically be a point in on the second line. And so any point shared on the first line will be automatically shared with the second line. And so that is the intersection of these two lines. Any solution to the first equation will be a solution to the second equation. And so for two-dimensional examples, we can see these are the only three possibilities. These are the only three ways that two lines can interact in the plane. They either intersect at a unique point. They either overlap and thus have an infinite intersection or they're parallel and they don't intersect each other ever. There's no solution.