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We're on problem 245. If x plus y is equal to a, and x minus y is equal to b, then what is-- they want to know what 2xy is equal to. Let's solve for x and y in terms of a and b, and then just figure out what this equals to. So we have two equations with two unknowns. Let's just add them together to solve for x. We get x plus x is 2x-- the y's cancel out-- is equal to a plus b. Or x is equal to a plus b over 2. Now if we had x plus y is equal to a-- I just re-wrote this. And now if we multiply both sides of this by negative 1, so we get minus x plus y is equal to minus b. And now add these two equations. So I'm essentially subtracting this equation from that one. That cancels out, so I get 2y is equal to a minus b, or y is equal to a minus b over 2. And now we can figure out what 2xy is equal to. 2xy is equal to 2 times a plus b over 2, times a minus b over 2. This 2 will cancel out with one of these 2's. And we're left with-- and what's a plus b times a minus b? It's a squared minus b squared over 2. Which is choice A. Problem 246. Let me do it in magenta. 246. A rectangular circuit board is to have width w inches-- let me draw it. So let's say it has width w inches, perimeter of p inches-- so let me just put that at the side right here-- perimeter of p inches, and area of k square inches. Which of the following equations must be true? And so they want us to relate this width to the area to the perimeter. Let me introduce another variable. Let's call this, right here, let's call this the height of the circuit board. So now we can do some interesting things. If that's the height, then that's also the height. If that's the width, then this is also the width. So let's see what the perimeter would be. It will be 2 times the width, plus 2 times the height is equal to the perimeter. And then we could also say width times height is equal to area. But if we can solve for height in terms of the perimeter and the width, then we could use that to get an expression that doesn't involve this variable. So let's do that. So if you divide both sides of this by 2, you get width plus height is equal to perimeter over 2. And then you get height is equal to perimeter over 2 minus width. And so the area, k, will be equal to the width times the height. Instead of writing an h there, let's write what we just figured out. p over 2 minus w. And then that is equal to pw/2 minus w squared. Let's see, when I look at the solution, they don't have any fractions in it, so let's multiple both sides of the equation by 2. We can ignore this. k is equal to pw/2 minus w squared. Multiply both sides by 2, you get 2k is equal to pw minus w squared. Let's add w squared the both sides. You get w squared plus 2k is equal to pw. Let's subtract, because all of the choices have them setting equal to 0. So then we could subtract pw from both sides, and you get-- oh wait, I made a mistake. If we multiply both sides of this by 2, 2 times k is 2k. 2 times pw/2 is pw. 2 times minus w squared is minus 2w squared. So in this step we have to add 2w squared to both sides. Sorry about that. So we have 2w squared plus 2k is equal to pw. Subtract pw from both sides, you get 2w squared minus pw, plus 2k is equal to 0. And that is choice E, right? 2w squared minus pw plus 2k. That is choice E. Next question. Problem 247. An arithmetic sequence is a sequence in which each term after the first is equal to the sum of the preceding term and a constant. If the list of letters shown above is an arithmetic sequence-- so they wrote p, r, s, t, u. So all they're saying is that, the difference between p and r is going to be some number. And the difference between r and s is going to be that same number. And the difference between s and t is going to be that same number. So an example of an arithmetic sequence, this could be 1, 2, 3, 4, 5. Because every number is one more than the one before it. So anyway, they say which of the following must also be an arithmetic sequence? So choice one, they write 2p, 2r, 2s, 2t, 2u. Well, let's just use our example. If this was p, r, s, t, and u, what is 2 times all of that? Well, then it'll be 2, 4-- no no, sorry-- it'll be, yeah, 2, 4, 6, 8, 10. So now instead of incrementing by one every time, we're incrementing by two every time. But it's still an arithmetic sequence because the difference between each number and the number before it is a constant. It's always equal to 2. So choice number one is definitely an arithmetic sequence. Choice two. And if you say, well, that works for that example, how do I know it'll work for all examples? So the other way to think about is, whatever the difference is between p and r, now you're going to have twice the difference between 2p and 2r. And it was the same distance between r and s, now you're going to have twice that distance between 2r and 2s. So here in our particular example, we went from 1 to 2, but it could have been something else. OK, Statement two says, p minus 3, r minus 3-- so they're just shifting everything; I don't even have to write them alll-- all the way to u minus 3. So they just took everything in this sequence and made them three less. But if r minus p is equal to some number, r minus 3, minus p minus 3 is going to be the exact same thing. I can even prove that to you, right? r minus p-- sorry, r minus 3, minus p minus 3. That's equal to r minus 3, minus p plus 3. And so these cancel out. So the difference between this and this ends up to be r minus p, the difference between that and that. And that should make sense intuitively, we're just shifting all the numbers down by three. So that shouldn't change the difference between the numbers. So two is still an arithmetic sequence. Statement three. p squared, r squared-- so we're just squaring all the numbers-- t squared, and u squared. So let's just use our particular example. If p was 1, then p squared is 1, 2 squared is 4, s squared is 9-- right, 3 squared is nine, 4 squared is 16. Now what's the difference between the numbers? The difference here is three. The difference here is five. The difference here is seven. And this is interesting in and of itself, that-- well, first of all, let's just answer our question. The difference is now not constant. We have a different difference between each successive number, right? At the beginning, the difference is two every time. Here it's three, then it changes to five, then it changes to seven. So three is not an arithmetic sequence. So the answer is D, one and two. This is something interesting. And if you've never experimented with it, this is something to look at. I've always been fascinated by the distance between the perfect squares increases by increasing odd numbers, which is just something to think about. Anyway, next problem. Actually, I have two problems left out of all of the problem-solving problems, so instead of just doing one problem and going over time, let me do two of them in the the next video. See you soon.