\(2^{x+1}.3^y=12^x\)
\(\Leftrightarrow2^{x+1}.3^y=\left(2^2.3\right)^x\)
\(\Leftrightarrow2^{x+1}.3^y=2^{2x}.3^x\)
\(\Leftrightarrow\left\{{}\begin{matrix}2^{x+1}=2^{2x}\\3^y=3^x\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x+1=2x\\y=x\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=1\\x=y\end{matrix}\right.\)
\(\Leftrightarrow x=y=1\)