\(\left\{{}\begin{matrix}\frac{1}{x}+\frac{1}{y}=\frac{3}{4}\\\frac{1}{2}\left(x+y\right)=3\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\frac{x+y}{xy}=\frac{3}{4}\\x+y=6\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\frac{6}{xy}=\frac{3}{4}\\x+y=6\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x+y=6\\xy=8\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y=6-x\\x\left(6-x\right)=8\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2-6x+8=0\\y=6-x\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x=4\\y=2\end{matrix}\right.\\\left\{{}\begin{matrix}x=2\\y=4\end{matrix}\right.\end{matrix}\right.\)
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