Worked example: evaluating expressions using structure

I'm now going to do a few classic problems that show up a lot at math competitions and sometimes on standardized tests, and they seem like very daunting and intimidating problems. But hopefully over the course of this video, you'll realize that if you kind of see what they're asking for, it's really not that bad. So let's try this first one right over here. They tell us a plus b plus c is equal to 7. And then they ask us, what is 5a plus 5b plus 5c equal to? And the first reaction's like, well, they just gave me one equation right over here. One equation with three unknowns, how do I solve for a, b, or c? Don't I have to be able to solve for a, b, or c in order to figure out what 5a plus 5b plus 5c is? So I'll let you think about that for a second. So the big idea here is to realize that 5a plus 5b plus 5c, that's just the same thing as 5 times a plus b plus c. If you distribute the 5 here, 5 times a, 5 times b, 5 times c, you get 5a plus 5b plus 5c. Another way you could think of it is we are factoring a 5 out. If you factor a 5 out, you're left with 5 times a plus b plus c. Now, how do we evaluate this? Well, that first equation gave us all the information that we needed. They told us that a plus b plus c is equal to 7. So 5 times a plus b plus c is the same exact thing. I think you see where this is going now. It's 5 times 7, and now this becomes pretty straightforward. That's going to be equal to 35. Let's do one more of these. And this is going to be a little bit more involved, but hopefully you see the same idea. So we're told a plus b plus c is equal to negative 1. And we have two more variables here. We're told that x plus y is equal to 7. Then they ask us this big, hairy thing. What is this equal to? And I'll give you a few seconds to give it a little bit of thought. Well, you could imagine. They've given three variables with one equation up here, another two variables with another equation. We have five unknowns with two equations. There's no way we're going to be able to individually figure out what a, or b, or c, or x, or y is. But maybe we can use what we saw in the last example to solve this. And so what we might want do is rearrange it so that we have the x's and the y's kind of grouped together, and the a's, b'c, and c's kind of grouped together. So let's try that out. So let's focus first on the a's, b's, and c's. So we have negative 9a, negative 9b-- I'll go in order, alphabetical order-- negative 9b, and we have negative 9c, and I think you see what's emerging. And then let's work on the x's and the y's. And then we have negative 7x, minus 7x, and then we have a minus 7y. So all I have done is rearranged this expression here. But this makes it a little bit clearer of what's going on here. These first three terms, I can factor out a negative 9, and I get negative 9 times a plus b plus c. And these second two terms, I can factor out a 7, minus 7 times x plus y. And just to verify, if you wanted to go the other way, multiply this negative 7 times x plus y, you'll get this. Multiply negative 9 times a plus b plus c, distribute it, you'll get this right over here. And so this makes it a little bit clearer. What is a plus b plus c equal to? Well, they tell us right over here. a plus b plus c is equal to negative 1. This whole expression is negative 1, at least in the parentheses. And what is x plus y equal to? Let me do this in a new color. What is x plus y equal to? Well, they tell us right over here. x plus y is equal to 7. So this whole thing simplifies to negative 9 times negative 1 minus 7 times 7. And so this is equal to-- we're in the home stretch-- negative 9 times negative 1. Well, that's positive 9. And then negative 7 times 7 is negative 49. So it's 9 minus 49, which is equal to negative 40. And we are done.