ГДЗ по алгебре 9 класс Макарычев Задание 512

\[\boxed{\text{512\ (512).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[\textbf{а)}\ \left\{ \begin{matrix} x^{2} + xy = 6 \\ y^{2} + xy = 3 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x^{2} + xy = 6\ \ \ \ \\ 2y^{2} + 2xy = 6 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x^{2} - xy - 2y^{2} = 0 \\ x^{2} - y^{2} = 3\ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} (x - 2y)(x + y) = 0 \\ x² - y^{2} = 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[1)\ \left\{ \begin{matrix} x = 2y\ \ \ \ \ \ \ \ \ \ \\ 4y^{2} - y^{2} = 3 \\ \end{matrix} \right.\ \Longrightarrow \left\{ \begin{matrix} x = 2y \\ y^{2} = 1\ \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} y_{1} = 1 \\ x_{1} = 2 \\ \end{matrix} \right.\ \text{\ \ \ }или\ \left\{ \begin{matrix} y_{2} = - 1 \\ x_{2} = - 2. \\ \end{matrix} \right.\ \]

\[2)\ \left\{ \begin{matrix} x = - y\ \ \ \ \ \ \ \\ y^{2} - y^{2} = 3 \\ \end{matrix} \right.\ \Longrightarrow \left\{ \begin{matrix} x = - y \\ 0 = 3\ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow корней\ нет;\]

\[\textbf{б)}\ \left\{ \begin{matrix} x^{2} - xy = 7 \\ y^{2} - xy = 9 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x - y = \frac{7}{x}\text{\ \ \ \ \ \ } \\ x - y = - \frac{9}{y} \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x - y = \frac{7}{x} \\ \frac{7}{x} = - \frac{9}{y}\text{\ \ \ \ } \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} y = - \frac{9}{7}x \\ x^{2} = \frac{49}{16}\text{\ \ \ } \\ \end{matrix} \right.\ \Longrightarrow \left\{ \begin{matrix} x = \pm 1,75 \\ y = - \frac{9}{7}\text{x\ \ } \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x_{1} = 1,75\ \ \ \\ y_{1} = - 2,25 \\ \end{matrix} \right.\ \text{\ \ \ }\]

\[или\ \left\{ \begin{matrix} x_{2} = - 1,75 \\ y_{2} = 2,25.\ \ \\ \end{matrix} \right.\ \]

\[Ответ:а)\ (2;1);( - 2;\ - 1);\]

\[\textbf{б)}\ ( - 1,75;2,25);(1,75;\ - 2,25)\text{.\ }\]