Topic A: Use properties of operations to generate equivalent expressions
I've gotten feedback that all the Chuck Norris imagery in the last video might have been a little bit too overwhelming. So for this video, I've included something a little bit more soothing. Let's try to simplify some more expressions. And we'll see we're just applying ideas that we already knew about. Let's say I want to simplify the expression 2 times 3x plus 5. Well, this literally means two 3x plus 5's. So this is the exact same thing. This is one 3x plus 5, and then to that, I'm going to add another 3x plus 5. This is literally what 2 times 3x plus 5 means. Well, this is the same thing as, if we just look at it right over here, we have now two 3x's. So we could write it as 2 times 3x. Plus, we have two 5's, so plus 2 times 5. You might say, hey, Sal, isn't this just the distributive property that I know from arithmetic? I've essentially just distributed the 2? 2 times 3x plus 2 times 5? And I would tell you, yes, it is. And the whole reason why I'm doing this is just to show you that it is exactly what you already know. But with that out of the way, let's continue to simplify it. When you multiply the 2 times the 3x, you get 6x. When you multiply the 2 times the 5, you get 10. So this simplified to 6x plus 10. Now let's try something that's a little bit more involved. Once again, really just things that you already know. Let's say I had 7 times 3y minus 5 minus 2 times 10 plus 4y. Let's see if we can simplify this. Well, let's work on the left-hand side of the expression, the 7 times 3y minus 5. We just have to distribute the 7. This is going to be 7 times 3y, which is going to give us 21y. Or if I had 3 y's 7 times, that's going to be 21 y's, either way you want to think about it. And then I have 7 times-- we've got to be careful with the sign-- negative 5. 7 times negative 5 is negative 35. So we've simplified this part of it. Let's simplify the right-hand side. You might be tempted to say, oh, 2 times 10 and 2 times 4y and then subtract them. And if you do that right and you distribute the subtraction, it would work out. But I like think of this as negative 2, and we're going to distribute the negative 2 times 10 and the negative 2 times 4y. So negative 2 times 10 is negative 20, so it's minus 20 right over here. And then negative 2 times 4, negative 2 times 4 is negative 8, so it's going to be negative 8y. Let's write a minus 8y right over here. And are we done simplifying? Well, no, there's a little bit more that we can do. We can't add the 21y to the negative 35 or the negative 20 because these are adding different things or subtracting different things. But we do have two things that are multiplying y. Let me do all in this green color. You have 21 y's right over here. And then we can view it as from that we are subtracting 8 y's. So if I have 21 of something and I take 8 of them away, I'm left with 13 of that something. So those are going to simplify to 13 y's. I'll do this in a new color. And then I have negative 35 minus 20. That's just going to simplify to negative 55. So this whole thing simplified, using a little bit of the distributive property and combining similar or like terms, we got to 13y minus 55.