I'm a senior in high school taking a number theory class. I have read the other answers on this site about back substitution but frankly I do not understand them at all. I know that back substitution is a very iterative process so I'm just looking for a simple explanation and a clear example because I feel like once I get it I will be able to solve all future problem with it.
Here's the problem I am trying to solve:
Use the GCD, along with back substitution, to obtain integers $x$ and $y$ that satisfy this equation: $\gcd(143,227)=143x+227y$.
I've already found the GCD using the Euclidean Algorithm and now just need to understand back substitution. I tried writing the first equation: $1=11-(5)(2)$, where $1$ is the last nonzero remainder obtained with the Euclidean Algorithm and $5$ and $2$ are the quotient and divisor respectively (from the same division problem). I understand that the next step is to substitute but I don't really understand how. This is where I'm at and any help is greatly appreciated.