Решебник по алгебре 9 класс Мерзляк Задание 991

\[\boxed{\mathbf{991\ (991).\ }Еуроки\ - \ ДЗ\ без\ мороки}\]

\[1\ ч\ 40\ мин = 1\frac{40}{60} = 1\frac{2}{3} = \frac{5}{3}\ ч.\]

\[Пусть\ x\ \frac{км}{ч} - скорость\]

\[\ одного,\ а\ y\ \frac{км}{ч} - скорость\ \]

\[другого.\]

\[Составим\ систему:\]

\[\left\{ \begin{matrix} 2 \cdot (x + y) = 20\ \ \ \ \ \ \ \ |\ :2 \\ \frac{20}{x} - \frac{20}{y} = \frac{5}{3}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x + y = 10\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 20y - 20x - \frac{5xy}{3} = 0 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\]

\[\text{\ \ }\left\{ \begin{matrix} x + y = 10\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 60y - 60x - 5xy = 0\ \ \ \ |\ :5 \\ \end{matrix} \right.\ \text{\ \ \ \ }\]

\[\text{\ \ \ \ }\left\{ \begin{matrix} x + y = 10\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 12y - 12x - xy = 0 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} x = 10 - y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 12y - 12 \cdot (10 - y) - y(10 - y) = 0 \\ \end{matrix} \right.\ \]

\[12y - 120 + 12y -\]

\[- 10y + y^{2} = 0\]

\[y^{2} + 14y - 120 = 0\]

\[y_{1} + y_{2} = - 14,\ \ \]

\[y_{1}y_{2} = - 120\]

\[y = - 20\ \ \ \ или\ \ \ \ y = 6\]

\[\left\{ \begin{matrix} y = 6 \\ x = 4 \\ \end{matrix} \right.\ \]

\[Ответ:6\ \frac{км}{ч},\ 4\frac{км}{ч}.\]