\(\left\{{}\begin{matrix}xy\left(x+2y\right)=3\\13x^3-4y^3=9\end{matrix}\right.\) (1)
\(\Rightarrow13x^3-4y^3=3xy\left(x+2y\right)\)
\(\Leftrightarrow13x^3-4y^3=3x^2y+6xy^2\)
\(\Leftrightarrow13x^3-3x^2y-6xy^2-4y^3=0\)
\(\Leftrightarrow x^3-3x^2y+3xy^2-y^3+12x^3-9xy^2-3y^3=0\)
\(\Leftrightarrow\left(x-y\right)^3+3\left(4x^3-3xy^2-y^3\right)=0\)
\(\Leftrightarrow\left(x-y\right)^3+3\left[\left(x^3-y^3\right)+3x^3-3xy^2\right]=0\)
\(\Leftrightarrow\left(x-y\right)^3+3\left[\left(x-y\right)\left(x^2+xy+y^2\right)+3x\left(x^2-y^2\right)\right]=0\)
\(\Leftrightarrow\left(x-y\right)^3+3\left[\left(x-y\right)\left(x^2+xy+y^2\right)+3x\left(x-y\right)\left(x+y\right)\right]=0\)
\(\Leftrightarrow\left(x-y\right)^3+3\left(x-y\right)\left(x^2+xy+y^2+3x^2+3xy\right)=0\)
\(\Leftrightarrow\left(x-y\right)^3+3\left(x-y\right)\left(4x^2+4xy+y^2\right)=0\)
\(\Leftrightarrow\left(x-y\right)^3+3\left(x-y\right)\left(2x+y\right)^2=0\)
\(\Leftrightarrow\left(x-y\right)\left[\left(x-y\right)^2+3\left(2x+y\right)^2\right]=0\)
\(\Leftrightarrow x-y=0\) hoặc \(\left\{{}\begin{matrix}x-y=0\\2x+y=0\end{matrix}\right.\) (2)
\(\left(2\right)\Leftrightarrow x=y=0\) thế vào (1) `=>` vô lý
\(\Leftrightarrow x=y\)
hệ \(\Rightarrow\left\{{}\begin{matrix}x^2.3x=3\\9x^3=9\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^3=1\\x^3=1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=1\\x=1\end{matrix}\right.\) `(tm)`
Vậy nghiệm hpt \(\left\{{}\begin{matrix}x=1\\y=1\end{matrix}\right.\)
