\(2^{x+1}\cdot3^y=12^x\)
\(\Rightarrow2^{x+1}\cdot3^y=\left(4\cdot3\right)^x\)
\(\Rightarrow2^{x+1}\cdot3^y=4^x\cdot3^x\)
\(\Rightarrow2^{x+1}\cdot3^y=2^{2x}\cdot3^x\)
\(\Rightarrow\left\{\begin{matrix}2^{x+1}=2^{2x}\\3^y=3^x\end{matrix}\right.\)
\(\Rightarrow\left\{\begin{matrix}x+1=2x\\x=y\end{matrix}\right.\)\(\Rightarrow x=y=1\)