 In this video I'm going to talk about solving linear equations. This is going to talk about identities and contradictions. So sometimes when we solve these equations some weird things happen and I'll show you kind of what those weird things mean. Alright, so here we go. I have an example of solving an equation. What I'm going to do is I'm going to just solve it like you normally would. Now this looks like a problem where I have, I got variables on both sides, I got variables on the left, I got v's on the left and a v on the right. So what I want to do is, okay I just want to solve it so I want to get all the v's to one side. Alright, so I'm going to start solving this. First thing on the left I see that I got v's here so I'm going to collect them together. Positive three, negative four makes a negative one v. Notice I don't put the one there, I just leave this negative v. Now the one is kind of redundant. Alright and on the right side I'm going to distribute that negative. This makes negative five minus v. Okay. Alright, now just like normal what I'm going to do is I'm going to make the decision, do I have my variables on the left or on the right? Let's just choose left. So what I want to do is I want to move all my variables to the left side. This one's on the left, I've got to move this one to the left side. So a negative v, I've got to add v to both sides, add v to both sides. Okay, so now what happens is that's zero, that's okay. So this equals negative five. But then this happens here. Negative v plus v gives me zero, so that actually goes away. And that's what's weird. I mentioned at the beginning of this video, sometimes weird things happen. And in this case, what happened is that the variables went away. What does that mean? Well in this case what that means is that, well we have two things that could mean it is either an identity or a contradiction, which one is it? Well look at what we just have here. Look at the bottom of what we've just solved. Negative nine is equal to negative five. Negative nine is equal to negative five. That is never true. This here is never, never true. Negative nine is never going to be equal to negative five. Those are two very, very different numbers. So that's never going to be true. So as you're solving these equations, if you get to something weird like this, you can actually use this weirdness to kind of evaluate what this is. If this is never true, this is what we call a contradiction. If this is never true, this is what we call a contradiction. So basically what this means is that there is no number. So here's a V, here's a V, here's a V. There is no single number that we can plug in to each one of these variables and get the same thing on the left side and on the right side. So basically means that no, that the variable is not equal to any number that we can find. So that's what a contradiction is. If you find something that's never true, it's a contradiction, which means we can't, there is no answer to it. Okay, so let's do another one. Let's do another example. 2 times the quantity x minus 6 equals negative 5x minus 12 plus 7x. There we go. Alrighty, so looking at this one, again, I'm going to solve this just like normal. And again, looking at this, it looks like I got variables on the left side, variables on the right side. So it looks like one of those type of problems. Again, I'm going to treat this like normal. I'm just going to start gathering things together that are alike. And I'm going to move the variable to where I deem necessary. So anyway, here we go. Alright, so I'm going to take the 2 and distribute it. So this is 2x minus 12 equals, alright, so negative 5x and a 7x. That makes 2x minus 12. Okay, now if you start looking at this, look at the left side and look at the right side. Back up a little bit, look at the left side and look at the right side. It's the exact same thing on the left and on the right side. Okay, so this is what we call an identity. And this is what we call an identity. If you see that everything on the left side and everything on the right side is happens to be the same, this is what we call an identity. What it means is that actually any variable is going to work. Anything is going to work. I can plug in the number 5. So if I put a 5 here, 2 times 5 is 10, 10 minus 12 is negative 2. 2 times 5 is 10, 10 minus 12 is negative 2. Negative 2, negative 2, it works. I can plug in whatever number I want to and it's always going to work. Okay, that's kind of the definition of identity. Alright, now the thing is, is not all students are going to see this right away. Okay, when you start solving it, you're just going to want to keep solving and keep solving. Alright, so if you want to keep solving, I'm just going to go through the normal process. So I have everything gathered, so now I want to decide, alright, let's get all the variables on the left side. That's seriously what students like to do. So let's subtract 2x, take this x and move it over to the other side. So subtract 2x, subtract 2x and what happens is all the x's go away. So now this kind of, this looks like a contradiction, but if you look at the very last, if you look at the very last line here, we have negative 12 is equal to negative 12. Negative 12 is equal to negative 12, which is actually very different from our first example. In this case, negative 12 is always equal to negative 12. This always happens. Negative 12 and negative 12 are the same number. Okay, so we can see the difference between the contradiction last time and the identity this time. Always true, that makes it an identity. And again, definition of identity is every number is going to work. What we also say in math land is that all real numbers are solutions. All real numbers are solutions. So again, you can plug in whatever you want to and you're going to get the same thing on the left side and on the right side. Okay, so that's how you find identities. That's how you find contradictions. And there's a couple of examples of both of them. It basically comes down to what you see at the very end. At the very end, is it always true? Then it's an identity. If it was never true, like our first example, if it's never true, then it's a contradiction.