ĐKXĐ: \(\left\{{}\begin{matrix}2x+y\ge0\\x-2y+1\ge0\end{matrix}\right.\).
Đặt \(\sqrt{2x+y}=a;\sqrt{x-2y+1}=b\left(a,b\ge0\right)\).
\(HPT\Leftrightarrow\left\{{}\begin{matrix}a+2b=5\\3b=b^2+a^2+1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}a+2b=5\\15b\left(a+2b\right)=25\left(b^2+a^2\right)+\left(a+2b\right)^2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}a+2b=5\\26a^2-11ab-b^2=0\end{matrix}\right.\).
\(\Leftrightarrow\left\{{}\begin{matrix}a+2b=5\\\left(2a-b\right)\left(13a+b\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}a+2b=5\\2a-b=0\end{matrix}\right.\) (Do 13a + b > 0)
\(\Leftrightarrow\left\{{}\begin{matrix}a=1\left(TM\right)\\b=2\left(TM\right)\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2x+y=1\\x-2y+1=4\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=-1\end{matrix}\right.\).
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