Germany|Math Olympiad|Algebra problem|solve for x and y.

Hi, let's solve for x and y. I want us to note that our x and y are natural numbers, okay? All right, now let's get straight to the point. Now when you have a squared minus b squared This is equal to a minus b multiplied by a plus b. Okay, now in this case, our a is x. while our b is equal to y, so that means that x squared minus y squared can be expressed as x minus y multiplied by x plus y, therefore the equation states x squared minus y squared. is equal to 20, which simply means that we can express the left-hand side of the equation if x minus y multiplied by x plus y equals 20, then we proceed accordingly. do this since we are told that x and y are natural numbers once they are natural numbers Then it simply means that x minus y is less than x plus y, okay, so that is the first thing. that we should all take note of now we can express 20 in so many ways 20 is equal to 1 multiplied by 20 20 multiplied by 1 2 multiplied by 10 10 multiplied by 2 then 4 multiplied by 5 and 5 multiplied by 4, so these are the possible ways we can express 20 All right, now since we say that x minus y must be less than x plus y, so if we say x minus y multiplied by x plus y means that the first number should be less than the second number. Right, therefore this is not going to be used, this is not going to be used, and this is not It is going to be used, yes, because here the first number is larger, which is not supposed to be. Here the first number is larger; it's not supposed to be like that, and also here the first number is bigger, it's not supposed to be like that, therefore we can now go ahead and solve. So we have for the The first values of x and y are such that 20 equals one multiplied by 20; we can now state that x minus y equals one, and x plus y equals 20, right? Yes, because we can simply say that x minus y multiplied by x plus y equals one multiplied by 20, so if we now Look at it very carefully; you can see that by comparison, x minus y equals one. and x plus y is 20, okay. Now let's solve: if we should add both equations together, x plus x All right, now minus y plus plus y gives us zero, which is equal to one plus 20, which is 21. Now to get The value of x, we divide this by two and this by two, and we have that x is equal to 21 over two. and this is what is not a real number, okay? Here, x is not natural, sorry, and therefore we're not I'm going to use this now, let's check the corresponding value of y. Okay, now if x is equal to 21 over 2. let's substitute 21 over 2 into any of these equations, so if we substitute it into the first We can say that 21 over 2 minus y is equal to 1, so we can write that 21 over 2 Then it is equal to 1 plus y, so we can say 21 over 2, this plus y, we move to the other side to become minus 1. is equal to y, all right? Now, if we solve this, we can see that we can write this as 21 over 2 minus 2 divided by 2 equals y, so we have 21 minus 2, which gives us 19 divided by 2 equals y, and this also holds true here. this is not what a natural number is; a natural number is not this, so we also reject this, so for The first values of x and y for which 20 equals 1 times 20 are those for which our x and y are Non-natural numbers, so they are rejected. Now let's check the second value for 20. equals 2 multiplied by 10; let's check this now. We are going to have that x minus y is equal to 2 and x plus y is equal to 10, okay. Now let's solve: if we add x plus x, we get 2x. Minus y plus y is 0, which is equal to 2 plus 10, which is 12. Now we can divide to get the value of x. so 2x divided by 2 and 12 divided by 2; now we can see this, and we can see that x is equal to 6 and 6 is a natural number, that's correct. Now that our x is equal to 6, let's substitute x equals 6. into any of these equations to verify that it is correct. Okay, sorry, to get the value of y, so if Substituting it into the first equation, we can see that 6 minus y equals 2, so 6 equals 2. plus y, then if we transfer this, we have 6 minus 2 equals y, and 4 equals y, so this is also a natural number, so this is also accepted. Now let's check the third one: for 20 equals Um, the third one is 20 equals 4 times 5, right? So for 20 equals 4 multiplied by 5. Okay, so we can see that x minus y equals 4 and x plus y equals 5. Please, I would like you to watch the end, yes, because I'm going to prove that the answer we got is right and you Know what the most interesting part of this video is? So please don't miss it for anything. Okay, all right. Now we can add the first and second equations: x plus x will give us 2x. and minus y plus y is 0, and 4 plus this is 9, okay. Now, to get the value of x, if we divide by 2. and divide by 2, we find that x will yield a fraction, which means that this is also It is not a real number, a natural number. Sorry, okay. Now, if our x is 9 over 2, let's substitute. into any of these equations to see if our y will be a natural number now let's go So we have x which is 9 over 2 minus y equal to 4, so 9 over 2 is equal to 4 plus y. Alright, so we have 9 over 2 minus 4 over 1 equals y, so we can write this as 9 over 2 minus 8 over 2 right is equal to y. Now when you look at this, you see 9 minus 8, which gives us 1. and this is over 2, which is equal to y, so you can still see that this is not a natural number, so This is also rejected; therefore, the only valid values for x and y are those where x equals 6. and y is equal to 4; now it's time to check. Okay, let's check. So if we check, the equation says that x squared minus y squared equals 20. Now record that our x is 6 and our y is 4, so This is what we have: 6, 4. So we have 6 squared minus 4 squared is equal to 20. Okay, now 6 squared. Is 36 minus 4 squared equal to 16? Is 36 minus 16 equal to 20? And is 20 equal to 20? So this It is actually correct, please click on the subscription button and turn on your notifications. like if this is the first time you are coming across this channel and please give us a thumbs up It means a lot to us if you actually enjoyed the video, which I'm sure you did. Thank you so much. For watching, bye