 So we've now learned how we can use the methods of substitution or elimination to solve a system of linear equations, at least if there's only two unknowns in the system. But when we started this lecture, lecture 11 here, we discussed that when it comes to the solution set to a linear system, sure, you can have a unique solution, like we've seen as examples when we've learned about elimination and substitution, but we've also seen geometrically that you can have no solution whatsoever, and it's also possible to have infinitely many solutions. Geometrically this made sense, but the methods of substitution or elimination are algebraic methods. How do you detect these geometric cases when you look at it only algebraically? That's what I want to talk about in this video here. Of course, I also want to introduce some appropriate terminology that we use when we describe the systems of equations. We say that a system of equations is consistent if it has a solution. Now, that could be any number of solutions. It could be one solution, it could be two solutions, it could be three solutions, it could be 5,280 solutions, it could be infinitely many solutions. If it has at least one solution, it's called consistent. And conversely, if it has no solution, then it's inconsistent. Now, when it comes to linear systems, there's only three possibilities, 0, 1, or infinity. So, if there are zero solutions, then you would be inconsistent. Now, if you're consistent for a linear system, that means you either have one or infinitely many solutions. The word consistent for a linear system does not distinguish between those two cases, but inconsistent does mean there are zero solutions. The one and infinity case, we will separate in just a moment, but let's now deal with this inconsistent case. How do you know when a system of equations is inconsistent? Well, it turns out that when you're solving a linear system, whether you're using substitution or elimination or other techniques that we will learn later on, a system, you'll discover that a linear system is inconsistent because you'll discover a contradiction. That is a statement which is definitely false. Like, it doesn't depend on a variable, it's false all of the time. This would be something like 0 equals 7. If you're solving your system of equations and you then discover that, oh, any solution to this system implies that 0 equals 7, that would give you a contradiction. Or, if you get something like 3 is equal to 5, these are contradictory statements. There's no choice of variable that can force a 0 to equal 7. Likewise, there's no choice of x or y that can make 3 become 5. Those are different integers, those are different real numbers. And so, if you discover a contradiction in the process of solving a linear system, that actually suggests that there is no solution, the system is inconsistent. So, consider the following linear system that you see here on the screen. Take 8x minus 2y is equal to 5 and negative 12x plus 2y is equal to 7. We can solve this system in one of two ways. We can do substitution or elimination that we've learned about in the previous videos for this lecture here. It doesn't really matter which one you use as long as you just pick one. And so, for this one, how about we solve it by, say, elimination? That seems like a pretty good method there. Like, if I want to eliminate the x's, like I noticed, for example, I have an 8x and I have a negative 12x. The x's have opposite signs already, so I just need to find the least common multiple between 8 and 12 and that's going to be 24. The LCM of 8 and 12 is 24, the least common multiple. 24 is 3 times 8, so I'm going to find times the first equation by 3 and 24 is 2 times 12, so I'm going to times the second equation by 2. This gives me the linear system of 24x minus 6y is equal to 15 and then for the second equation, you're going to get negative 24x. 2 times 3y is going to give you positive 6y and then on the right hand side, you're going to get 2 times 7, which is 14, like so. And so, as I combine these equations together, a curious thing happens. Now, as I expected, the 24x adds to the negative 24x and they cancel each other out. I was looking for that, I was trying to eliminate x, but the curious thing that happens here is that as I try to eliminate the x's, I simultaneously eliminate the y's. So, you have a negative 6y plus 6y, those are going to cancel out and on the left hand side, you end up with a 0. Now, that's not immediately a call for alarm, but whenever you get something like this, when the entire left hand side cancels out, this is going to be suggestive that we do not have a unique solution. It's going to be one of the other two cases, either 0 solutions or infinitely many solutions. Now, on the right hand side, you get 15 plus 14 that adds up to, of course, to be 29 and you get this contradiction. You get that 0 is equal to 29, no it ain't, that's not a thing. And therefore, because we get this contradiction, this would tell us that there is in fact no solution to this linear system. The only way there could be a solution is if 0 equals 29, which is not the case. And so, this is an example of this inconsistent case that we were talking about just a moment ago. And so, as you describe this system, you can say that there's no solution. You can write it out, no solution. Sometimes people use this symbol, it's like a little circle with a slash through it. This would suggest that the solution set is empty, so that means no solution. And for these linear systems, we describe this as inconsistent. Feel free to use some type of, one of these notations here to denote that there is no solution that it's an inconsistent system. Now, you can't just say the system's inconsistent. I mean, a student would need to show their work. And the work you show is that you need to find a contradiction. If you derive a contradiction, then you can declare that the linear system is inconsistent. If you just say it's inconsistent without any evidence whatsoever, even if you're right, it's like, uh, was that a guess? We know it's inconsistent because we found this contradiction. Now, let's look at the flip side, right? We've considered, since there's three possibilities, 0, 1, and infinity, we've seen the 0 and 1 case. How do you get infinitely many solutions in a linear system? Well, like I mentioned before, a system is consistent if it has a solution. For a linear system, that means there could be one solution or infinitely many solutions. How do you distinguish between the two cases? Well, again, we've seen how to find unique solutions. What indicates that you have infinitely many solutions? Now, a little bit of terminology again here. If a system of equations has a finite number of solutions, then the equations are going to have to be independent. Now, for a linear system, that would be because there's a unique solution in there. Likewise, if the system has infinitely many solutions, that would make the equations dependent, okay? Now, I should mention that as we're talking about linear systems, this is very much like the birth of linear algebra, which for SU course numberings, that's math 2270. In which case these terms, I'm not being super precise with them right now. In a linear algebraic setting, we would be much more precise with them, but for a college algebra setting, these are appropriate understandings of the terms here. We say that the equations are independent if you have a unique solution, and we say that the equations are dependent if they have infinitely many solutions. Now, how do you discover if we're in the dependent case? Because that basically gives us the three cases we're looking for. There's the inconsistent case, there's the independent case, which have a unique solution, and there's the dependent case, where you have infinitely many solutions. To detect the dependent case, it's actually very similar to the inconsistent case. But this time, instead of looking for a contradiction, we're looking for a tautology, an identity, so to speak. We're looking for a statement that's always true, regardless of the assignment of the variables. So this could be something like 0 equals 0. Regardless of what you choose for x and y, 0 equals 0, that tautology tells us that we actually have infinitely many solutions. If you get 2 equals 2, right, 7 equals 7, these are statements as they're universally true, they're going to allow for more than one solution to this linear system. And when a linear system has two solutions, you can actually generate infinitely many solutions from there. Because any linear combination of solutions will create a new solution. Well, not any linear combination, but again, I'm getting too much into the math 2270 stuff and getting too far away from the math 1050. If you have this statement, these universally true statements like 0 equals 0, if you're doing elimination, you're always going to get 0 equals 0. If you're doing substitution, you might get something else like 2 equals 2. If you discover one of these things, when you're solving a system of linear equations, you're going to get dependent equations. Okay, meaning there's more than one solution. And we'll actually describe what the general solution looks like in that situation. So let's look at this one right here. We already solved one by elimination. Let's try to solve one by substitution. After all, when I look at the first equation, 3x minus 6y is equal to 12. I notice that everyone in the first equation is divisible by 3. If you divide by 3, then you're going to get x minus 2y is equal to 4. And then 4x minus 8y equals 16. I'm not going to touch that one right now. I can then easily solve the first equation for x, right? If I just move the 2y to the other side, we're going to get that x equals 4 plus 2y. I didn't want to substitute that in to the other equation for x there. So if I do that, we're going to get 4 times 4 plus 2y minus 8y. That's then going to equal 16. Simplifying the left-hand side, I'm going to distribute the 4. So we end up with 16 plus 8y minus 8y is equal to 16. If I add the y's together, you get 8y minus 8y. They cancel out, and you end up with 16 equals 16. Whoa, this is exactly what we were looking for. This indicates to us that we actually are in the dependent case. So some things to note here is that we're in the dependent case. And again, the terminology independent versus dependent, one learns exactly why we use these terms in linear algebra. But we're just in proto-linear algebra right now, so we'll use those terms nonetheless. There's going to be infinitely many solutions in this situation. But honestly, it's not good enough to say there's infinitely many solutions, because not every point in the plane is a solution, even though there's infinitely many. So what we actually do is to kind of suggest where this name comes from, we actually are going to choose one of our variables to be a dependent variable. So there's these two variables in play here. There's x and y. It turns out that y is going to be a deep, excuse me, an independent variable. Sometimes it's called a free variable, meaning that, ah, independent. It's got to finish writing this out. It's a free variable. So when it comes to our solution set, it turns out that y can be any number you want. Any number under the sun. And so as such, she sometimes introduced a new symbol, maybe like a t, to suggest that y could be anything. So y is equal to t. Then since x, taken this equation from before, since x is equal to 4 plus 2y, since y is equal to some number t, x we can then compute to be 4 plus 2t. And so this then leads to the so-called general solution, the general solution to this system of equations. You get the following. x looks like 4 plus 2y, and y looks like, sorry, well, I mean, x does look like 4 plus 2y, but given that y is equal to some number t, we get that x is equal to 4 plus 2t. Like so we have this new, this new parameter. So t is just some arbitrary filler. It's just a free number we can put in there. And so as you pick different values of t, you'll get different solutions. Like if t was equal to 0, you plug 0 into this formula here, you're going to end up with 4 comma 0. That is, in fact, a solution to this linear system. If t equals 1, you're going to end up with 6 comma 1. If t equals 2, you end up with 8 comma 2. If t equals negative 1, you'd get the point 2 comma negative 1. And so this is why we get infinitely many solutions. The solutions are, there's infinitely many because we have a free variable in the system. And because we have this free variable, we can freely choose it to be any real number we want. And as if there's infinitely many real numbers, we can get infinitely many distinct solutions. And so that then exhausts the three possibilities you get for a system of linear equations with two equations, two unknowns. You can get a unique solution, as we've seen before. You can get no solution, the inconsistent case, or you can get infinitely many solutions for which we can describe the general solution to find all those infinitely many solutions. That is the dependent case. And all three of these cases can be found using either the methods of substitution or elimination. So that then finishes lecture 11 for us. We'll talk some more about this in lecture 12. If you learned anything about linear systems of equations, please like these videos, subscribe to the channel to see more videos like this in the future. If you have friends or colleagues who could benefit from these videos, feel free to share them with them. And finally, of course, if you have any questions whatsoever, please don't hesitate to post them in the comments below and I'll be glad to answer them at my soonest convenience.