\(\left\{{}\begin{matrix}\left(m-3\right)x+y=2\\mx+2y=8\end{matrix}\right.\rightarrow\left\{{}\begin{matrix}2\left(m-3\right)x+2y=4\\mx+2y=8\end{matrix}\right.\)
\(\rightarrow\left\{{}\begin{matrix}mx-\left(2m-6\right)x=4\\\left(m-3\right)x+y=2\end{matrix}\right.\)
\(\rightarrow\left\{{}\begin{matrix}\left(6-m\right)x=4\left(1\right)\\y=2-\left(m-3\right)x\end{matrix}\right.\)
\(\left(1\right)\rightarrow\left\{{}\begin{matrix}6-m\ne0\\4⋮\left(6-m\right)\end{matrix}\right.\)
\(\rightarrow\left\{{}\begin{matrix}m\ne6\\\left(6-m\right)\in\left\{-4;-2;-1;1;2;4\right\}\end{matrix}\right.\)
\(\rightarrow m\in\left\{10;8;7;5;4;2\right\}\)
\(\rightarrow x\in\left\{-1;-2;4;;4;2;1\right\}\)
\(\rightarrow y\in\left\{9;12;18;-6;0;3\right\}\)