I've got a linear diophantine equation to solve over $\mathbb{Z}/17\mathbb{Z}$. The equation is part of a linear system with 3 unknowns and 3 equations. However solving this system through gaussian elimination over $\mathbb{Z}/17\mathbb{Z}$ does not result in all possible solutions, and this equation only contains 2 of the unknowns of that linear system, so it should be solvable on its own. I've tried everything I can think of, but I've been unable to come up with all solutions so I hope you can help!
Here's the equation,
$$15x + 10y = 4 \pmod{17}$$
I've further reduced it to
$$8x + 11y = 1 \pmod{17}$$
but that doesnt seem to help me too much either. I tried representing it as follows
$$8x + 11y + 17z = 1$$ and then trying to solve the equation, however this doesnt seem to get me any closer to a solution, so I'm quite stuck. Perhaps I'm missing something. WolframAlpha gives the solution as $y = 7(n+2)$ and $x = n$.
Any help is greatly appreciated!