A Nice Entry Exam Question | Can You Solve?

Added: 2026-05-08

[00:00:00]Hello everyone. Happy to see you here.

[00:00:01]Welcome back to my channel Hi am a mathematics. Today we have great algebra question x squared minus y squared equal to 36. We need to solve this question for x and y. So if you have your solution, your answer you can also write it in the comments below. It will be really interesting to read about it.

[00:00:17]Okay, so first of all on the left side we have difference of squares. This is our old known formula, so let's start with that. So we have x plus y x plus y times x minus y x minus y equal to 36.

[00:00:32]Right now if you have thoughts about the 36 because on the left side we have a product of two parentheses. We have a product of two expressions. And so right now let's try to express this 36 as a product of two constant, yeah? Because the first way how can we express this 36 is 36 times one. Maybe the the the basic one, yeah?

[00:00:52]One times 36 is also a great expression because this is equal to 36.

[00:00:57]Uh 12 times three is the next one. 12 times three. Three times 12. Three times 12. Uh nine times four. Nine times four.

[00:01:06]Four times nine. Four times nine. And 18 times two.

[00:01:11]And two times 18. So we have a lot of expression. And so right now a few really interesting thoughts about it because x plus y is greater than x minus y. So we can consider our left constant constant a greater than the the right one. This is is greater than this one.

[00:01:28]So as a result 36 times one is good expression for us. One times 36 is not good. 12 times three is good for us.

[00:01:37]Three times 12 we we reject it.

[00:01:40]Uh nine times four we good for us. Four times nine we reject it. 18 times two is good and two times 18 we we reject it.

[00:01:48]Okay, so right now let's write all these these combinations. So the first combination in our case uh we have x plus y x plus y times x minus y equal to 36 times one. This is our first first system of equations.

[00:02:05]Second combination x plus y times x minus y x minus y is equal to uh 12 times three. This is our second expression 12 times three.

[00:02:18]Third third combination x plus y x plus y times x minus y x minus y equal to nine times four. Nine times four.

[00:02:31]And fourth combination x plus y x plus y times x minus y equal to 18 times two. 18 times two. So we don't need to check all of this. We check only these four four combinations. So well right now let's start uh let's start and let's solve our first first system of equations. So from here x plus y equal to 36. So x plus y equal to 36 and x minus y equal to one. X minus y equal to one. How can we solve it? This is a basic system of two equations. We just need to add this. When we add it we cancel this y. So 2x equal to 37.

[00:03:15]So 2x equal to 37. From here x is equal to 37 over over two.

[00:03:24]And let's solve it for y. Let's plug in this x right here. So 37 over two minus y equal to equal to one. Let's multiply both side by two because we don't need this uh fraction right here. So 37 minus 2y equal to equal to two. As a result from here our minus 2y is equal to minus 35.

[00:03:52]And from here our y is equal to 35 over over two. This is our first first pair. So we have 37 half.

[00:04:03]37 half. And the second one 35 half. 35 half. Of course this is not the integer pair, but that don't matter. We solve the first system of equation. This is the solution to this to our to our question. Okay, let's go to the next one. Second system of equations 12 and three. So we have x plus y equal to 12.

[00:04:23]x plus y equal to 12.

[00:04:27]And x minus y x minus y equal to equal to three. Okay, so let's use the same logic. Let's add these two equations. So 2x equal to 15.

[00:04:40]And from here x is equal to 15 15 half.

[00:04:46]And right now let's plug in instead of this x let's plug in 15 half. So we have 15 half.

[00:04:51]15 half minus y right here minus y equal to three. So as a result when we multiply it by two we have 15 minus 2y 15 minus 2y equal to six. So minus 2y equal to minus nine. And y equal to nine.

[00:05:12]Nine half. So our second pair of roots 15 half.

[00:05:18]And nine half from here. Nine half. So we want to underline that this is our first and second second root. And of course let's go to the third. Let's go to the third system of equations. Okay, let's do it. What do we have? Third one nine and four. Nine and four. So the third system of equations x plus y equal to nine.

[00:05:42]And x minus y equal to equal to four. So let's use the same logic. Let's add it. So 2x equal to 13.

[00:05:53]2x equal to 13. And from here x equal to 13 half. Equal to 13 13 half. Okay, let's go to the next step. Let's plug in it instead of this x let's plug in 13 half and let's solve for y. So 13 half minus y equal to four. Multiplying by two.

[00:06:14]And we have 13 minus 2y equal to eight.

[00:06:20]Minus 2y minus 2y equal to minus five.

[00:06:26]And y equal to five half.

[00:06:29]Five half. Okay, so let's write our third uh third pair. So 13 half.

[00:06:36]And five half.

[00:06:38]And five half. Okay, we have three pairs. And our last step 18 and two. Our last system of equations. So we have 18 half.

[00:06:47]18 sorry 18 and two. So x plus y equal to 18.

[00:06:53]And x minus y equal to equal to two.

[00:06:58]Okay, really great. So right now let's add everything. So we have 2x equal to 20. And from here x is equal to is equal to 10. This is really great because this is our integer integer root. So let's plug in it right here. So 10 minus y so 10 minus y is equal to is equal to two. So 10 y equal to equal to eight. So we have our integer integer pair. So we have 10 and eight. This is our fourth fourth pair. So we let's write our final answer and let's check real quick our our roots. So our answer to this question our answer First of all we have an integer pair. So I'm going to start it I'm going to start with it. So 10 and eight.

[00:07:43]Second pair we have 37 half. 37 half.

[00:07:48]And 35 half. 35 half.

[00:07:51]Third pair 15 half.

[00:07:54]And nine half.

[00:07:56]And nine half. And the last pair we have uh 13 half five half we we have it. Yeah, we have 13 half and five half. 13 half.

[00:08:06]And five half. Okay, this is our our solution. This is non integer. We we're not talking about This is non integer solution. Integer.

[00:08:19]And this is our integer solution. We are talking about natural numbers 10 and and eight. And in the end let's check real quick. So our check we have x squared minus y squared equal to 36. Let's check real quick our 10 and eight. So 10 squared minus eight squared equal to 30 36. So 100 minus 64 equal to 36. Which is absolutely correct because 36 is equal to 36. So our root is absolutely great. You know sometimes happen that you need to solve this question only for natural numbers. So that's why you need to solve it like that. But a lot of times you need to solve this question completely. So that's why you need you don't need to forget about uh this another branch. We have different different roots. We have our first, second, and third. So this is a full solution. You know sometimes happen that a lot of student they solve this question by inspection because they see that 100 minus 64 equal to 36. This is also really good, but just agree with me this is extremely weird solution if you solve this question like that. So first line, second, and third. And agree with me that this is the better way to solve this question. Just you just write your combinations on the right side. And of course eliminate something, reject something, and just solve this four system of equation. And in the end you can easily say okay, I solved this question completely and step by step.

[00:09:52]Okay, so in our question our correct answer is 10 and eight. Also you can use this. This variations of solution you can also check it. And yeah, this is my solution to this question. I really hope you understand it. I really hope you learn something new, but definitely don't feel bad if you got this wrong. If you need help with any of these classes, I have a lot of questions on my YouTube channel, a lot of different challenges, so it's also really interesting and it's really great you to see it, to to check it on on on this channel. And I want to say thank you everyone for being here, for watching it. It's extremely I'm extremely happy about it and I want to say thank you for for for being here and a lot of teachers, a lot of students here and it's also really great because my main goal is to make math clear and understandable for everyone and that's why I really appreciate it that you write a comment, write your thoughts, write your respond in the comments below. And of course, the main thing is when you like understand my solution, when you ask something, when you learn something, this is I'm really happy about it.

[00:10:52]So, thank everyone for your time. Take care of yourself. Have a great day. See you in the next videos.