Đáp án:
\[\left[ \begin{array}{l}
x = y = 0\\
x = y = 1\\
x = y = - 1
\end{array} \right.\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\left\{ \begin{array}{l}
{x^3} = 2y - x\\
{y^3} = 2x - y
\end{array} \right.\\
\Leftrightarrow {x^3} - {y^3} = \left( {2y - x} \right) - \left( {2x - y} \right)\\
\Leftrightarrow \left( {x - y} \right)\left( {{x^2} + xy + {y^2}} \right) = 3\left( {y - x} \right)\\
\Leftrightarrow \left( {x - y} \right).\left( {{x^2} + xy + {y^2} + 3} \right) = 0\\
{x^2} + xy + {y^2} + 3 = \left( {{x^2} + xy + \frac{1}{4}{y^2}} \right) + \frac{3}{4}{y^2} + 3 = {\left( {x + \frac{1}{2}y} \right)^2} + \frac{3}{4}{y^2} + 3 > 0,\,\,\,\forall x;y\\
\Rightarrow x - y = 0\\
\Leftrightarrow x = y\\
{x^3} = 2y - x\\
\Leftrightarrow {x^3} = 2x - x\\
\Leftrightarrow {x^3} - x = 0\\
\Leftrightarrow x\left( {x - 1} \right)\left( {x + 1} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
x = y = 0\\
x = y = 1\\
x = y = - 1
\end{array} \right.
\end{array}\)
