 In this video I'm going to be talking about properties of equality, there's four different properties here and there's more on a separate page that I'm going to go over. Addition property of equality, subtraction property of equality, multiplication property of equality and also the division property of equality. Now it's a little bit repetitive with the properties of equality and we're just going over the four that are here. These are the four operation properties, you can call them that. I'm just going to go over kind of in general what they look like and then try to relate them to something that you already know. I'm going to use a conditional statement to say this. Conditional statement is an if-then statement. So if A is equal to B, then A plus C is equal to B plus C. So this addition property of equality, what it means is if I have an equivalent statement, so here's my equal sign, equivalent statement, if I have this statement, then if I add something to one side of the statement, then I have to add that same thing to the other side, notice the plus C and the plus C. Notice that those are there. If I add something to one side, I must add it to the other. Now that should sound really familiar. You hear that all the time from your math teacher when you start solving equations. Whenever I add something to one side of the equal sign, I must add it to the other side. So these are properties that are basically used when we solve equations. So all of these properties we have already seen, what we're doing is we are just taking these properties and writing them down as rules, as properties. We're taking these things that you already know and we're kind of defining them. That's one thing that we do in mathematics, we define everything. Okay, so as you can well imagine, this is the subtraction property of equality. If A is equal to B, then A minus C is equal to B minus C. So again, the subtraction property is that you have to whatever you subtract from one side, you must subtract that same thing from the other side of the equation. Okay, again, something that you've seen when solving equations. Okay, multiplication property, again, so you can well imagine that we're going to multiply if, and if I can, sorry about that, I wrote that down wrong. So if A is equal to B, then if I multiply something on both sides, I can say C times A is equal to C times B. Now notice I put the C out front. It doesn't matter if the C is out front or if it's behind, we can do AC is equal to BC. That also works. Either way, it doesn't matter. But if you multiply something on one side of the equation, you must multiply it on the other side. Okay, so now for the division property of equality, again, using my conditional if then statement, if A is equal to B, then A over C, A divided by C is equal to B divided by C. So notice that C I've been using the entire time is that thing I'm either adding, subtracting, multiplying, or dividing on each side. Now there is one special rule I have to go over with the division property, and I also have to say that C, little comma here, C cannot equal zero. Now think about that for a moment. C can't equal zero. Now why not? Well, if we look at our then portion, we look at our conclusion. If C is zero, then we are dividing by a negative number, which is a big no-no. You can't divide by a negative number. It doesn't matter what the equations look like. Can't divide by zero. So a lot of times, especially when you get into your higher, higher level of mathematics, you have to have small little rules like this because sometimes you encounter things that you simply can't do. In this case, we've encountered we cannot divide by zero. Okay, so moving on to the next properties. These ones, properties of equality continued. Okay, so now we're going over the reflexive property of equality, symmetric property of equality, transitive property of equality, and also the substitution property of equality. Now these four properties, a little bit more complicated than the first ones, but I'll go over them briefly. Okay, so the reflexive property of equality kind of in the same style as the other ones, a is equal to a. A number is always going to be equal to itself. That's what the reflexive property is. A number is always going to be equal to itself. Pretty simple property. Symmetric property on their hand is when a little bit more complicated. If a is equal to b, then b is equal to a. Okay, so that basically is just switching it around. If we have a equality statement, the one side of the equality statement is equal to the other side, well then the other side is then going to be equal to the first part. Is basically how that kind of reads. Not overly complicated. These are just very, very basic ones, but again, these symmetric properties are really, really useful. Once we start doing algebraic proofs. Okay, so then last or second to last year, transitive property. This one's a little bit more complicated, a little bit more involved. If a is equal to b, and b is equal to c, then a is equal to c. Okay, now in a previous lesson that I did, we talked about the law of syllogism. This is actually very similar to that law of syllogism. Law of syllogism is if a bear is a wild animal, and wild animals are dangerous, then bears are going to be dangerous. That's kind of what we use for the law of syllogism. It's kind of the same thing here, except we're just using this with numbers. Using this with variable expressions instead of story problems. So that's basically what the transitive property is. And then the substitution property. The substitution property. Now, there's a number of different ways to do the substitution property. I can either go over a regular example. I can either go over a regular example, or I can do kind of a formal definition. I think I'm going to do just a little bit of both. I'm going to do a little bit of both. There's a number of different ways to do the substitution property. But I think what I'll do is kind of just a little bit of both using the variables and numbers, and kind of doing a formal definition. What I like to do, use for substitution, is replacing a term with something that is equivalent. Equivalent. Okay, replacing a term with something that's equivalent. So whenever you take something out and then replace it back in, that's the substitution property of equality. You've seen a lot, since you're in this level of mathematics, you've seen a lot of substitution when you solve equations, or when you plug something back into an equation, whatever that is. Whenever you take something out and put something else in, that's the same thing, that's equivalent. That's what we call a substitution. So for example, one thing that I could say is if x is equal to 9, then 3x is equal to, if I take this and kind of plug it in, if x is equal to 9, 3 times 9 is going to be 27. That's kind of one form of the substitution property. If I take this 9 and plug it in here, I will get 27. That's one way to think of it. There's lots of different examples that you can do. Not one example is going to kind of cover them all. All right. Anyway, those are the properties of equalities. I'm hoping that was helpful for you. There's the addition property. Oh, let me go back. Let me go back. We have the addition property of equality, subtraction property of equality, multiplication property of equality, division property of equality, and then we have the reflexive property of equality. Symmetric property of equality, transitive property of equality, and the substitution property of equality. And those are all the properties, a lot of the properties that you're going to use that you're going to be seeing, either in your algebra level courses, maybe algebra two in your geometry level courses.