What's the technique used by this teacher to solve ax+by=GCD(a,b) in RSA Algorithm-Cryptography?

It starts at 6:16



It's the part where he's pointing his hand in this picture. I didn't get it although it's pretty mechanical, so I'd like to learn that technique as this is really useful in RSA algorithm(rather than having to memorize some values). I've been searching for a method like that since I saw this problem(been months) but could not find a technique like that. And it was always about just do it rather having any mechanical method like this one. So, I want to learn this.


Or, if you have any other simple methods to solve this equation, please tell me