Решебник по алгебре 9 класс Мерзляк Задание 467

\[\boxed{\text{467\ (467).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[1)\ \left\{ \begin{matrix} x + y - xy = 1 \\ x + y + xy = 9 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x + y - xy + x + y + xy = 10 \\ x + y - xy - x - y - xy = - 8 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} 2x + 2y = 10\ \ \ \ \ |\ :2 \\ - 2xy = - 8\ \ \ \ \ \ \ \ \ \ \ |\ :2 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} x + y = 5 \\ xy = 4\ \ \ \ \ \\ \end{matrix} \right.\ \]

\[x = 5 - y\]

\[5y - y^{2} - 4 = 0\]

\[y_{1} + y_{2} = 5,\ \ y_{1} = 4\]

\[y_{1}y_{2} = 4,\ \ y_{2} = 1\]

\[\left\{ \begin{matrix} y = 4 \\ x = 1 \\ \end{matrix} \right.\ \ \ \ или\ \ \ \left\{ \begin{matrix} y = 1 \\ x = 4 \\ \end{matrix} \right.\ \]

\[Ответ:(1;4),\ (4;1).\]

\[2)\ \left\{ \begin{matrix} 3xy + 2x = - 4 \\ 3xy + y = - 8\ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} 3xy + 2x - 3xy - y = 4 \\ 3xy + 2x = - 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} 2x - y = 4\ \ \ \ \ \ \ \ \\ 3\text{xy} + 2x = - 4 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} y = 2x - 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 3x \cdot (2x - 4) + 2x + 4 = 0 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} y = 2x - 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 6x^{2} - 12x + 2x + 4 = 0\ \ \ \ |\ :2 \\ \end{matrix} \right.\ \]

\[3x^{2} - 5x + 2 = 0\]

\[D = 25 - 24 = 1\]

\[x_{1,2} = \frac{5 \pm 1}{6}\]

\[\left\{ \begin{matrix} x = 1\ \ \ \\ y = - 2 \\ \end{matrix} \right.\ \ \ \ \ или\ \ \ \ \left\{ \begin{matrix} x = \frac{2}{3}\text{\ \ \ \ \ } \\ y = - \frac{8}{3} \\ \end{matrix} \right.\ \]

\[Ответ:(1;\ - 2);\ \left( \frac{2}{3};\ - \frac{8}{3} \right).\]

\[3)\ \left\{ \begin{matrix} xy - x = 24 \\ xy - y = 25 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} xy - x - xy + y = - 1 \\ xy - y = 25\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} y - x = - 1 \\ xy - y = 25 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} y = x - 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x(x - 1) - x + 1 - 25 = 0 \\ \end{matrix} \right.\ \]

\[x^{2} - x - x + 1 - 25 = 0\]

\[x^{2} - 2x - 24 = 0\]

\[x_{1} + x_{2} = 2,\ \ x_{1} = 6\]

\[x_{1}x_{2} = - 24,\ \ x_{2} = - 4\]

\[\left\{ \begin{matrix} x = 6 \\ y = 5 \\ \end{matrix} \right.\ \ \ \ \ или\ \ \ \left\{ \begin{matrix} x = - 4 \\ y = - 5 \\ \end{matrix} \right.\ \]

\[Ответ:(6;5),\ ( - 4;\ - 5).\]

\[4)\ \left\{ \begin{matrix} 2x^{2} + y^{2} = 66 \\ 2x^{2} - y^{2} = 34 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} 2x^{2} + y^{2} - 2x^{2} + y^{2} = 32 \\ 2x^{2} + y^{2} + 2x^{2} - y^{2} = 100 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} 2y^{2} = 32 \\ 4x^{2} = 100 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ }\left\{ \begin{matrix} y^{2} = 16 \\ x^{2} = 25 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} y = \pm 4 \\ x = \pm 5 \\ \end{matrix} \right.\ \]

\[Ответ:(5;4),\ (5;\ - 4),\ ( - 5;4),\ \]

\[( - 5;\ - 4)\text{.\ }\]