Решите систему уравнений: x-y+xy=-4; xy(x-y)=-21.

Ответ:

\[\left\{ \begin{matrix} x - y + xy = - 4 \\ \text{xy}(x - y) = - 21 \\ \end{matrix} \right.\ \]

\[Пусть\ (x - y) = t;\ \ xy = c:\]

\[\left\{ \begin{matrix} t + c = - 4 \\ ct = - 21\ \ \ \\ \end{matrix} \right.\ \text{\ \ }\]

\[По\ теореме,\ обратной\ \]

\[теореме\ Виета:\]

\[t = - 7,\ \ \ c = 3\ \ или\ \ c = - 7,\ \ \ \]

\[t = 3.\]

\[y_{1,2} = \frac{7 \pm \sqrt{61}}{2}\]

\[x_{1,2} = \frac{7 \pm \sqrt{61} - 14}{2} = \frac{- 7 \pm \sqrt{61}}{2}\]

\[Ответ:\left( \frac{- 7 - \sqrt{61}}{2};\ \frac{7 - \sqrt{61}}{2} \right);\ \ \]

\[\left( \frac{- 7 + \sqrt{61}}{2};\ \frac{7 + \sqrt{61}}{2} \right).\]