Решебник по алгебре 9 класс Макарычев Задание 512

 

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

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

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

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

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

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

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

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

\[\Longrightarrow корней\ нет;\]

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

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

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

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

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

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

\[Ответ:а)\ (2;1);( - 2;\ - 1);\]

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

\[\boxed{\text{512.}\text{\ }\text{ОК\ ГДЗ\ -\ домашка\ на\ 5}}\]

\[Пусть\ \text{x\ }\frac{км}{ч} - скорость\ \]

\[первого\ поезда,\ y - второго.\]

\[Составим\ систему\ уравнений:\]

\[\left\{ \begin{matrix} x + y = \frac{270}{3}\text{\ \ \ \ \ \ \ \ \ \ } \\ \frac{270}{x} = \frac{270}{y} + 1\frac{21}{60} \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x + y = 90\ \ \ \ \ \ \ \ \\ \frac{270}{x} - \frac{270}{y} = \frac{81}{60} \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} y = 90 - x\ \ \ \ \ \ \ \ \ \ \ \ \\ \frac{10 \cdot (y - x)}{\text{xy}} = \frac{1}{20} \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} y = 90 - x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 200 \cdot (90 - 2x) = 90x - x^{2} \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} y = 90 - x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - 490x + 18000 = 0 \\ \end{matrix} \right.\ \]

\[D = 245^{2} - 18000 = 42025 =\]

\[= 205^{2}\]

\[x_{1,2} = 245 \pm 205;\]

\[\left\{ \begin{matrix} x_{1} = 40 \\ y_{1} = 50 \\ \end{matrix} \right.\ \ \ \ \ \ или\ \]

\[\ \left\{ \begin{matrix} x_{2} = 450\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y_{2} = 90 - 450 < 0 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow не\ подходят.\]

\[Ответ:40\ \frac{км}{ч}\ и\ 50\ \frac{км}{ч}.\]