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So I think I found these $$27y + 23 = 32x$$ $$81y + 85 = 128x$$ in a text-book or something, and it was a graphing problem. (These are not simultaneous equations, they are separate.)
I tried to find integer solutions to this and after putting in some numbers I still couldn't find any.
I used some graphing software and still could not find any integer solutions for $x, y \in \Bbb Z$.
So I wonder, do any solutions exist? But more importantly, is there a technique for checking if equations like these actually have integer solutions, if so what is the technique.
This isn't overly important, but if there was such a technique that would be helpful.