Do there exist whole number solutions to $27y + 23 = 32x$ and $81y + 85 = 128x$? 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. Thank you.

By the **Bézout identity** , these two equations do have solutions.

The criterion for existence is that the $\gcd$ of the coefficients of $x$ and $y$ must divide the constant term.

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By the way,

$$27\cdot11+23=32\cdot10,$$

$$81\cdot59+85=128\cdot38.$$