ГДЗ по алгебре 9 класс Макарычев Задание 448

\[\boxed{\text{448\ (448).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[\textbf{а)}\ \left\{ \begin{matrix} x^{2} - 2y^{2} = 14 \\ x^{2} + 2y^{2} = 18 \\ \end{matrix} \right.\ ( + )\]

\[\left\{ \begin{matrix} 2x^{2} = 32\ \ \ \ \ \ \ \ \ \\ 2y^{2} = x^{2} - 14 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x^{2} = 16\ \ \ \ \ \ \ \ \ \\ y^{2} = \frac{x^{2} - 14}{2} \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} x_{1} = 4 \\ y_{1} = 1 \\ \end{matrix} \right.\ \ \ \ \ или\ \ \ \left\{ \begin{matrix} x_{2} = 4\ \ \\ y_{2} = - 1 \\ \end{matrix} \right.\ \ \ или\]

\[\ \left\{ \begin{matrix} x_{3} = - 4 \\ y_{3} = 1\ \ \ \\ \end{matrix} \right.\ \ \ \ или\mathbf{\text{\ \ \ }}\left\{ \begin{matrix} x_{4} = - 4 \\ y_{4} = - 1 \\ \end{matrix} \right.\ .\]

\[\textbf{б)}\ \left\{ \begin{matrix} x^{2} + y^{2} = 61 \\ x^{2} - y^{2} = 11 \\ \end{matrix} \right.\ ( + )\]

\[\left\{ \begin{matrix} 2x^{2} = 72\ \ \ \ \ \ \ \ \\ y^{2} = x^{2} - 11 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x^{2} = 36\ \ \ \ \ \ \ \ \ \ \\ y^{2} = 36 - 11 \\ \end{matrix} \right.\ \Longrightarrow \left\{ \begin{matrix} x^{2} = 36 \\ y^{2} = 25 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} x_{1} = 6 \\ y_{1} = 5 \\ \end{matrix} \right.\ \ \ или\ \ \left\{ \begin{matrix} x_{2} = 6\ \ \ \\ y_{2} = - 5 \\ \end{matrix} \right.\ \ \ или\ \]

\[\ \left\{ \begin{matrix} x_{3} = - 6 \\ y_{3} = 5\ \ \ \ \\ \end{matrix} \right.\ \ \ \ или\ \ \ \left\{ \begin{matrix} x_{4} = - 6 \\ y_{4} = - 5 \\ \end{matrix} \right.\ .\]

\[\textbf{в)}\ \left\{ \begin{matrix} xy + x = 56 \\ xy + y = 54 \\ \end{matrix} \right.\ \ ( - )\]

\[\left\{ \begin{matrix} x - y = 2\ \ \ \ \ \ \ \ \\ y(x + 1) = 54 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x = y + 2\ \ \ \ \ \ \\ y(y + 3) = 54 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x = y + 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y^{2} + 3y - 54 = 0 \\ \end{matrix} \right.\ \]

\[y^{2} + 3y - 54 = 0\]

\[y_{1} + y_{2} = - 3;\ \ \ y_{1} \cdot y_{2} = - 54\]

\[y_{1} = - 9;\ \ \ y_{2} = 6.\]

\[\Longrightarrow \left\{ \begin{matrix} y_{1} = - 9 \\ x_{1} = - 7 \\ \end{matrix} \right.\ \ \ \ или\ \ \ \left\{ \begin{matrix} y_{2} = 6\ \\ x_{2} = 8. \\ \end{matrix} \right.\ \]

\[Ответ:а)\ (4;1);(4;\ - 1);\]

\[( - 4;1);( - 4;\ - 1);\]

\[\textbf{б)}\ (6;5);(6;\ - 5);\]

\[( - 6;5);( - 6;\ - 5);\]

\[\textbf{в)}\ ( - 7;\ - 9);(8;6).\]