Equal and equivalent expressions

0:00- [Lecturer] All right, what we need to do for each of these statements, we need to compare these expressions. And in this blank, we either need to put an always, a sometimes, or never. So why don't we pause this video and have a go at that before we do it together?

0:13Okay, now let's look at this first one. So 7x minus 4x plus five, how does that compare to 3x plus five? Well, my impulse is, let me try to simplify what I have here on the left. So if I add or if I take 7x and I subtract 4x's from that, I'm going to be left with 3x.

0:30So this is going to become 3x plus five. Well, for any x, 3x plus five is always equal to 3x plus five. They're the same thing. So I'll write always, always, right over there. All right, let's do this next one. Nine y minus 3y plus four and we have all this business on the right.

0:51Well, let's simplify both of these. If I combine 9y and this negative 3y, I get 6y, and then I still have that plus four there. Here, if I have 6y minus 2y, I have a 4y and I'm going to have plus 12. So these aren't identical expressions. These are different. So it's definitely not always, but could there be some time where these are going to be the same, maybe for a particular y value?

1:21And I'd argue yes. In fact, you could solve for that y value. You could set them equal to each other and figure out when they are the same. So let me do that actually. You don't have to do that, but just to prove to you, 6y plus four is equal to 4y plus 12. Well, if you subtract 4y, let me actually, oh, it's gonna get a little messy, but let me just do it.

1:42If I subtract 4y from both sides, what am I going to have? I'm going to get 2y, 6y minus 4y is 2y, plus four. I know that's hard to see. Let me write it over here. Two y plus four is equal to, and then I'm going to have, these are going to cancel out, and I'm just going to be left with a 12.

2:03And then if I subtract four from both sides, this is still messy. I should have probably done it someplace else. I'm going to get these cancel out. I'm going to be left with 2y on the left and that's going to be equal to eight. And then if you divide both sides by two, you would get y is equal to eight over two, which is equal to four.

2:21So there definitely is a time where these two expressions can be equal to each other. That's when y is equal to four. So, but it's not for any y, only for y equals four. So I would write sometimes, sometimes. Last but not least, these two over here. Well, let's see.

2:40Once again, I'm going to simplify this. So 6y minus y is 5y and I have 5y plus eight. And then here I have 5y minus eight. So could I have something plus eight ever be equal to that same something minus eight? No, it doesn't matter what y is, and I could actually even try to solve it.

3:03I could say 5Y plus eight is equal to 5y minus eight. If you subtract 5y from both sides, which is a legitimate thing to do, I'm going to get eight is equal to negative eight, which is impossible. There's no y value that can make that true. So this is definitely a never, but just reason through it.

3:28There's no scenario where if I, whatever this is, if I add eight, it could be equal to whatever that same thing is minus eight.