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Video transcript

Which expressions are equivalent to 2 times the quantity 4f plus 2g? Mark all that apply. So the first one here is 8f plus 4g. So could I manipulate this somehow, in a valid way, so I get 8f plus 4g? Well, the most obvious thing is I just distribute the 2. So 2 times this whole quantity is going to be 2 times 4f plus 2 times 2g, which is equal to 8f plus 4g. So this expression is indeed equivalent to that expression right over there. Now let's see what they tried to do. 2f times this thing right over here. Is that equivalent to that over there? Well, it doesn't look like it's going to be. And you could even try to distribute this right over here. You're going to get-- if we distribute the 2f, you'll get 2f times 4 is 8f. And 2f times 2g is plus 4fg, which is very different than 8f plus 4g. This is 8f plus 4fg. So this one is not an equivalent expression. And here that you have 8f plus 2g. Well, 8f plus 2g, we already know, is different than 8f plus 4g, And 8f plus 4g is an equivalent expression. And these two things aren't equivalent, so we can cross that out. Now they have a 4 times 2f plus g. Well, what happens if we were to factor a 4 out of 8f plus 4g, which we already know is equivalent to our original expression? So if you try to factor out a 4 right over here, so you divide 8 by 4, you get 2f. And you divide 4g by 4, you get g. And you can't just only divide by 4. Then you then have to multiply by 4 in order to not change the actual value of the expressions. So all we did is we divided by 4 and then multiplied by 4, which doesn't change the actual value. Or you could think of it as we undistributed a 4. We factored out a 4. So 4 times 2f plus 4g is indeed the same thing as 8f plus 4g, which is, we already know, is the same thing as our original expression. And you can distribute this to verify that. So this is also a valid expression. Let's do a couple more of these. Fill in the blank to produce an expression equivalent to mu plus mu plus mu. Well, I have 3 mus right over here. So this is literally just going to be 3 mu, 3 times the variable mu. That's all that's going on right over there. Let's do a couple more of this. That was pretty fast. Which expressions are equivalent to 6l plus 5m minus 3n? Mark all that apply. So let's look at this first one. If I were to distribute the 3, I would get 6l minus 3n plus 5. And that and that are equivalent. If you just change the order-- oh, a plus 5m. Let me be very careful here. So if I were to distribute the 3, 6l minus 3n plus 5m-- and this expression and this expression are going to be equivalent if you just swap the negative 3m and the 5m, which you could completely do. Addition is commutative. It doesn't matter which order you actually add in. So this is legitimate. Now let's see. 3n plus 6l. So already, something goes shady here. Here you have a minus 3n, or you could view it as a negative 3n. Here you have a positive 3n, and they don't fix it anywhere else. So this does not seem like a legitimate expression. And they also have a negative 5m, while it was a positive 5m right over there. So that's definitely not the case. So here you have 5m, a positive 5m. Well, you have a positive 5m right over there. Then you have plus 6l minus 3n. Putting these parentheses, these are essentially reassociating what operation I would do first. But you could actually remove the parentheses here, and it won't change the value. You could think about it as distributing the positive sign or distributing the positive 1 here. It would just become 5m plus 6l minus 3n, which is just a reordering of this right over here. But it's completely legitimate. So let me write that right over there. And then they have 5l. And if you distribute this, this would be 3m. So they mixed everything up. This is 6l. This is a 5l. So this one right over here is no good. This is a lot of fun. Let's keep going. If we take the expression 2 times a plus 2b-- in parentheses-- and ignore the parentheses, we can write another expression, 2a plus 2b, if you were to do that. Is 2a plus 2b equivalent to 2 times a plus 2b in parentheses? So when they're saying all of this thing over here, they're saying, hey, look, I'm just this irresponsible mathematician. And I like to just ignore parentheses without thinking about it fully. And if I did, I would just get 2a plus 2b. Can I do this? And you might already be imagining, based on the tone of my voice, what the answer is. And to think about it, you just have to realize, well, look. You're multiplying 2 not just times the a. You're multiplying 2 times the entire quantity a plus 2b. You have to distribute that 2. This is going to be equal to 2 times a plus 2 times 2b, which is equal to 2a plus 4b. So you can't just ignore that there. 2a plus 4b is very different than 2a plus 2b. So are these two things equivalent? No. No, this is seriously irresponsible mathematics.