Here is an interesting solution to a two-variable one-equation system.
We want to solve .
Subtracting from both sides, we see that .
Subtracting from both sides of the original equation, we get that .
Now we have a system of equations:
Plugging the second equation into the first, we get that .
Now, plugging this equation into itself recursively, we see that
To solve , we observe the following sequence:
Agree that the limit of this sequence as it goes on to its infinite term is .
Also, note that this sequence is equivalent to the following sequence:
As we all know from various proofs which I will not outline here (this sequence is pretty famous), the limit of this sequence is .
However, this limit is equivalent to , as we know that .
Therefore, our solution to our original system of equation is , and we are done.