# Can't seem to solve linear diophantine equation over $\mathbb{Z}/17\mathbb{Z}$.

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!

• I noticed that, in $\Bbb Z/(17\Bbb Z)$ we have$$15x+10y=4\iff -2x+10y=4\iff 5y=2+x\iff y=7(2+x)$$since $5\cdot 7=35=1$. – Dave Nov 2 '17 at 12:30
• Or just render $x=5y-2$ in the last step. – Oscar Lanzi Nov 2 '17 at 12:58