\(taco:\left\{{}\begin{matrix}x^3+y^3=1\\x^2y+2xy^2+y^3=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left(x+y\right)\left(x^2-xy+y^2\right)=1\\y\left(x^2+2xy+y^2\right)=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left(x+y\right)\left(2x^2-2xy+2y^2\right)=2\\y\left(x+y\right)^2=2\end{matrix}\right.\Rightarrow\left(x+y\right)\left[2x^2-2xy+2y^2-y\left(x+y\right)\right]=0\Leftrightarrow\left(x+y\right)\left(2x^2-2xy+2y^2-xy-y^2\right)=0\Leftrightarrow\left(x+y\right)\left(2x^2-3xy+y^2\right)=0\Leftrightarrow\left(x+y\right)\left[\left(2x^2-xy\right)-\left(2xy-y^2\right)\right]=0\Leftrightarrow\left(x+y\right)\left[x\left(2x-y\right)-y\left(2x-y\right)\right]=0\Leftrightarrow\left(x+y\right)\left(x-y\right)\left(2x-y\right)=0\) \(\Leftrightarrow\left[{}\begin{matrix}x+y=0\\x-y=0\\2x-y=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-y\\x=y\\x=\frac{1}{2}y\end{matrix}\right.\)
\(+,x=-y\Rightarrow x^3=\left(-y\right)^3=-y^3\Rightarrow x^3+y^3=-y^3+y^3=0\ne1\left(loại\right)\) \(+,x=y\Rightarrow\left\{{}\begin{matrix}x^3+y^3=1\\x^2y+2xy^2+y^3=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2x^3=1\\x^3+2x^3+x^3=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2x^3=1\\4x^3=2\end{matrix}\right.\Leftrightarrow x^3=\frac{1}{2}\Leftrightarrow x=\sqrt[3]{\frac{1}{2}}\Rightarrow x=y=\sqrt[3]{\frac{1}{2}}\) \(+,x=\frac{1}{2}y\Rightarrow\left\{{}\begin{matrix}x^3+y^3=1\\x^2y+2xy^2+y^3=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\frac{9}{8}y^3=1\\\frac{1}{4}y^3+\frac{2.1}{2}y^3+y^3=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\frac{9}{8}y^3=1\\\frac{9}{4}y^3=2\end{matrix}\right.\Leftrightarrow y^3=\frac{8}{9}\Leftrightarrow y=\sqrt[3]{\frac{8}{9}}\Rightarrow x=y=\sqrt[3]{\frac{8}{9}}\)
\(Vậy:\left(x,y\right)\in\left\{\left(\sqrt[3]{\frac{1}{2}};\sqrt[3]{\frac{1}{2}}\right);\left(\sqrt[3]{\frac{8}{9}};\sqrt[3]{\frac{8}{9}}\right)\right\}\)
mk nhầm TH 3 phai là:
\(x=\sqrt[3]{\frac{8}{9}}=\frac{2}{\sqrt[3]{9}}\Rightarrow y=\frac{4}{\sqrt[3]{9}}\)
nha
