Решебник по алгебре 9 класс Мерзляк Задание 464

\[\boxed{\text{464\ (464).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

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

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

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

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

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

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

\[3y^{2} - 3y = 0\]

\[3y \cdot (y - 1) = 0\]

\[y = 0\]

\[y = 1\]

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

\[Ответ:(1;0),\ (0;1).\]

\[2)\ \left\{ \begin{matrix} x^{3} - y^{3} = 28\ \ \ \ \ \ \ \ \\ x^{2} + xy + y^{2} = 7 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} (x - y)\left( x^{2} + xy + y^{2} \right) = 28 \\ x^{2} + xy + y^{2} = 7\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} 7x - 7y = 28\ \ \ \ \ \ \ \\ x^{2} + xy + y^{2} = 7 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x - y = 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} + xy + y^{2} = 7 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = 4 + y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ (4 + y)^{2} + y \cdot (4 + y) + y^{2} = 7 \\ \end{matrix} \right.\ \]

\[16 + 8y + y^{2} + 4y + y^{2} +\]

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

\[3y^{2} + 12y + 9 = 0\ \ \ \ \ \ \ \ \ \ |\ :3\]

\[y^{2} + 4y + 3 = 0\]

\[y_{1} + y_{2} = - 4,\ \ y_{1} = - 3\]

\[y_{1}y_{2} = 3,\ \ y_{2} = - 1\]

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

\[Ответ:(1;\ - 3),\ (3;\ - 1).\]

\[3)\ \left\{ \begin{matrix} x^{2} - y^{2} = 7 \\ xy = 12\ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x^{2} - y^{2} = 7 \\ x = \frac{12}{y}\text{\ \ \ \ \ \ \ \ \ \ } \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} \left( \frac{12}{y} \right)^{2} - y^{2} = 7 \\ x = \frac{12}{y}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \end{matrix} \right.\ \]

\[\frac{144}{y^{2}} - y^{2} = 7\]

\[144 - y^{4} - 7y^{2} = 0,\]

\[\ \ y \neq 0\]

\[Пусть\ y^{2} = t,\]

\[{- t}^{2} - 7t + 144 = 0,\ \ t \geq 0\]

\[t_{1} + t_{2} = - 7,\ \ \]

\[t_{1} = - 16 - не\ удовлетворяет\]

\[t_{1}t_{2} = - 144,\ \ t_{2} = 9\]

\[y^{2} = 9\]

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

\[Ответ:(4;3),\ ( - 4;\ - 3).\]

\[4)\ \left\{ \begin{matrix} 3x^{2} - 2y^{2} = 19 \\ xy = - 6\ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} 3x^{2} - 2y^{2} = 19 \\ x = - \frac{6}{y}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} 3 \cdot \left( - \frac{6}{y} \right)^{2} - 2y^{2} = 19 \\ x = - \frac{6}{y}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \end{matrix} \right.\ \]

\[\frac{108}{y^{2}} - 2y^{2} - 19 = 0;\ \ \ \ y \neq 0\]

\[108 - 2y^{4} - 19y^{2} = 0\]

\[Пусть\ y^{2} = t,\ тогда:\]

\[- 2t^{2} - 19t + 108 = 0,\ \ \]

\[t \geq 0\]

\[2t^{2} + 19t - 108 = 0\]

\[D = 361 + 864 = 1225\]

\[t_{1,2} = \frac{- 19 \pm 35}{4}\]

\[t_{1} = 4\]

\[t_{2} = - \frac{27}{2} - не\ удовлетворяет.\]

\[y^{2} = 4\]

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

\(Ответ:( - 3;2);\ (3;\ - 2).\)