Решебник по алгебре 9 класс Мерзляк Задание 984

\[\boxed{\mathbf{984\ (984).\ }Еуроки\ - \ ДЗ\ без\ мороки}\]

\[1)\left\{ \begin{matrix} x - 4y = - 6 \\ x^{2} + 4y^{2} = 8 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\text{\ \ }\left\{ \begin{matrix} x = 4y - 6\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ (4y - 6)^{2} + 4y^{2} = 8 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} x = 4y - 6\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 16y^{2} - 48y + 36 + 4y^{2} = 8 \\ \end{matrix} \right.\ \]

\[20y² - 48y + 28 = 0\ \ |\ :4\]

\[5y^{2} - 12y + 7 = 0\]

\[D = 144 - 140 = 4\]

\[y = \frac{12 + 2}{10} = 1,4;\ \]

\[\text{\ \ \ \ }y = \frac{12 - 2}{10} = 1\]

\[\left\{ \begin{matrix} y = 1,4\ \ \ \ \ \ \ \ \ \ \ \ \\ x = 4 \cdot 1,4 - 6 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\left\{ \begin{matrix} y = 1,4\ \ \\ x = - 0,4 \\ \end{matrix} \right.\ \ \]

\[\ \ \ \ или\ \ \ \ \left\{ \begin{matrix} y = 1\ \ \ \ \ \ \\ x = 4 - 6 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\left\{ \begin{matrix} y = 1\ \ \ \\ x = - 2 \\ \end{matrix} \right.\ \]

\[Ответ:( - 0,4;1,4);\ \ ( - 2;1).\]

\[2)\ \left\{ \begin{matrix} 3x + y = - 2\ \ \ \ \ \\ 3x^{2} - 2xy = 28 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} y = - 2 - 3x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 3x^{2} - 2x( - 2 - 3x) = 28 \\ \end{matrix} \right.\ \]

\[3x^{2} + 4x + 6x^{2} - 28 = 0\]

\[9x² + 4x - 28 = 0\]

\[D = 16 + 1008 = 1024\]

\[x = \frac{- 4 + 32}{18} = \frac{14}{9},\]

\[\ \ x = \frac{- 4 - 32}{18} = - 2\]

\[\left\{ \begin{matrix} x = - 2\ \ \ \ \ \ \\ y = - 2 + 6 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\left\{ \begin{matrix} x = - 2 \\ y = 4\ \ \ \\ \end{matrix} \right.\ \ \ \ или\ \ \]

\[\ \left\{ \begin{matrix} x = \frac{14}{9}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ y = - 2 - \frac{3 \cdot 14}{9} \\ \end{matrix} \right.\ \text{\ \ \ \ }\]

\[\text{\ \ }\left\{ \begin{matrix} x = \frac{14}{9}\text{\ \ \ \ \ \ \ \ \ \ } \\ y = - 2 - \frac{14}{3} \\ \end{matrix} \right.\ \text{\ \ \ \ }\]

\[\ \left\{ \begin{matrix} x = \frac{14}{9}\text{\ \ \ \ } \\ y = - \frac{20}{3} \\ \end{matrix} \right.\ \]

\[Ответ:( - 2;4),\ \left( \frac{14}{9};\ - \frac{20}{3} \right).\]

\[3)\ \left\{ \begin{matrix} x + 2y = 13 \\ xy = 15 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = 13 - 2y \\ y(13 - 2y) = 15 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} x = 13 - 2y \\ 13y - 2y^{2} - 15 = 0\ \ | \cdot ( - 1) \\ \end{matrix} \right.\ \]

\[2y^{2} - 13y + 15 = 0\]

\[D = 169 - 120 = 49\]

\[y = \frac{13 + 7}{4} = 5,\ \ \ \ \]

\[\ \ y = \frac{13 - 7}{4} = \frac{3}{2}\]

\[\left\{ \begin{matrix} x = 13 - 2 \cdot 5 \\ y = 5 \\ \end{matrix} \right.\ \text{\ \ \ \ }\left\{ \begin{matrix} x = 3 \\ y = 5 \\ \end{matrix} \right.\ \ \ \ \ \ или\ \ \ \ \]

\[\left\{ \begin{matrix} x = 13 - \frac{2 \cdot 3}{2} \\ y = \frac{3}{2}\text{\ \ \ \ \ \ \ \ } \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\left\{ \begin{matrix} x = 10 \\ y = \frac{3}{2} \\ \end{matrix} \right.\ \]

\[Ответ:(3;5);\left( 10;\frac{3}{2} \right)\text{.\ }\]

\[4)\ \left\{ \begin{matrix} 3x - 2y = 18 \\ xy = - 12 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} x = \frac{18 + 2y}{3} \\ y \cdot \left( \frac{18 + 2y}{3} \right) = - 12\ \ \ | \cdot 3 \\ \end{matrix} \right.\ \]

\[y(18 + 2y) = - 36\]

\[18y + 2y² + 36 = 0\ \ \ |\ :2\]

\[y^{2} + 9y + 18 = 0\]

\[D = 81 - 72 = 9\]

\[y = \frac{- 9 + 3}{2} = - 3,\]

\[\ \ y = \frac{- 9 - 3}{2} = - 6\]

\[\left\{ \begin{matrix} x = \frac{18 - 6}{3} \\ y = - 3 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\left\{ \begin{matrix} x = 4 \\ y = - 3 \\ \end{matrix} \right.\ \ \ \ \ или\ \]

\[\ \left\{ \begin{matrix} x = \frac{18 - 12}{3} \\ y = - 6 \\ \end{matrix} \right.\ \text{\ \ \ \ }\left\{ \begin{matrix} x = 2 \\ y = - 6 \\ \end{matrix} \right.\ \]

\[Ответ:(4;\ - 3),\ (2;\ - 6).\]

\[5)\ \left\{ \begin{matrix} x^{2} - y^{2} = 21 \\ x + y = - 3 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} ( - 3 - y)^{2} - y^{2} = 21 \\ x = - 3 - y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} 9 + 6y + y^{2} - y^{2} = 21 \\ x = - 3 - y \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} 6y = 12 \\ x = - 3 - y \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\left\{ \begin{matrix} y = 2 \\ x = - 3 - 2 \\ \end{matrix} \right.\ \text{\ \ \ }\]

\[\text{\ \ \ \ \ }\left\{ \begin{matrix} y = 2 \\ x = - 5 \\ \end{matrix} \right.\ \]

\[Ответ:( - 5;2).\]

\[6)\ \left\{ \begin{matrix} x^{2} - xy + y^{2} = 7 \\ x - y = 1 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} (1 + y)^{2} - y(1 + y) + y^{2} = 7 \\ x = 1 + y \\ \end{matrix} \right.\ \]

\[1 + 2y + y^{2} - y - y^{2} +\]

\[+ y^{2} - 7 = 0\]

\[y^{2} + y - 6 = 0\]

\[(y + 3)(y - 2) = 0\]

\[y = - 3,\ \ y = 2\]

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

\[\ \left\{ \begin{matrix} x = 1 + 2 \\ y = 2 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\left\{ \begin{matrix} x = 3 \\ y = 2 \\ \end{matrix} \right.\ \]

\[Ответ:( - 2;\ - 3),\ (3;2).\]

\[7)\ \left\{ \begin{matrix} (x - 1)(y - 1) = - 2 \\ x + y = 1 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} (1 - y - 1)(y - 1) = - 2 \\ x = 1 - y \\ \end{matrix} \right.\ \text{\ \ \ }\]

\[\ \left\{ \begin{matrix} - y(y - 1) = - 2 \\ x = 1 - y \\ \end{matrix} \right.\ \]

\[- y^{2} + y + 2 = 0\ \ \ | \cdot ( - 1)\]

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

\[(y - 2)(y + 1) = 0\]

\[y = 2,\ \ y = - 1\]

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

\[\ \left\{ \begin{matrix} x = 1 + 1 \\ y = - 1 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\left\{ \begin{matrix} x = 2 \\ y = - 1 \\ \end{matrix} \right.\ \]

\[Ответ:( - 1;2),\ (2;\ - 1).\]

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

\[\left\{ \begin{matrix} (3 + y - 2)(y + 1) = 1 \\ x = 3 + y \\ \end{matrix} \right.\ \text{\ \ \ }\]

\[\ \left\{ \begin{matrix} (1 + y)^{2} = 1 \\ x = 3 + y\ \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} 1 + y = 1 \\ x = 3 + y \\ \end{matrix} \right.\ \text{\ \ \ \ }\left\{ \begin{matrix} y = 0 \\ x = 3 \\ \end{matrix} \right.\ \ \ \ \ \ или\ \ \ \ \]

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

\[Ответ:(3;0),\ (1;\ - 2).\]

\[9)\ \left\{ \begin{matrix} \frac{1}{x} + \frac{1}{y} = \frac{3}{8} \\ x + y = 12 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\left\{ \begin{matrix} \frac{y + x}{\text{xy}} = \frac{3}{8} \\ x + y = 12 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} \frac{12}{\text{xy}} = \frac{3}{8} \\ x = 12 - y \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\left\{ \begin{matrix} 3xy = 96\ \ |\ :3 \\ x = 12 - y \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} xy = 32 \\ x = 12 - y \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\left\{ \begin{matrix} (12 - y)y = 32 \\ x = 12 - y\ \ \\ \end{matrix} \right.\ \text{\ \ \ \ }\]

\[\ \left\{ \begin{matrix} 12y - y^{2} - 32 = 0 \\ x = 12 - y \\ \end{matrix} \right.\ \]

\[y^{2} - 12y + 32 = 0\]

\[(y - 8)(y - 4) = 0\]

\[y = 8,\ \ y = 4\]

\[\left\{ \begin{matrix} y = 8 \\ x = 12 - 8 \\ \end{matrix} \right.\ \text{\ \ \ \ }\left\{ \begin{matrix} y = 8 \\ x = 4 \\ \end{matrix} \right.\ \ \ \ \ или\ \ \ \]

\[\ \left\{ \begin{matrix} y = 4 \\ x = 12 - 4 \\ \end{matrix} \right.\ \text{\ \ \ }\left\{ \begin{matrix} y = 4 \\ x = 8 \\ \end{matrix} \right.\ \]

\[Ответ:(4;8),\ (8;4).\]

\[10)\ \left\{ \begin{matrix} \frac{1}{x} - \frac{1}{y} = \frac{4}{5} \\ y - x = 4 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ }\left\{ \begin{matrix} \frac{y - x}{\text{xy}} = \frac{4}{5} \\ y - x = 4 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

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

\[\left\{ \begin{matrix} \frac{1}{\text{xy}} = \frac{1}{5} \\ y = 4 + x \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\left\{ \begin{matrix} xy = 5 \\ y = 4 + x \\ \end{matrix} \right.\ \text{\ \ \ }\]

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

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

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

\[x_{1}x_{2} = - 5,\ \ x_{2} = 1\]

\[\left\{ \begin{matrix} x = - 5 \\ y = - 1 \\ \end{matrix} \right.\ \ \ \ \ \ \ \ или\ \ \ \ \ \ \ \left\{ \begin{matrix} x = 1 \\ y = 5 \\ \end{matrix} \right.\ \]

\[Ответ:( - 5;\ - 1),\ (1;5).\]