\[t,\ ч\] | \[V,\ \frac{км}{ч}\] | \[S,\ км\] | |
---|---|---|---|
\[І\ авто\] |
\[\frac{420}{x + 20}\] \[на\ 2\ ч\ 24\ мин < \searrow \ \] |
\[x + 20\] | \[420\] |
\[ІІ\ авто\] | \[\frac{420}{x}\ \] | \[x\] | \[420\] |
\[24\ ч\ 24\ мин = 2\ ч\frac{24}{60} = 2\ ч\frac{2}{5} = 2\frac{2}{5}\ ч.\]
\[Составим\ уравнение:\]
\[\frac{420}{x} - \frac{420}{x + 20} = 2\frac{2}{5}\]
\[\frac{420 \cdot (x + 20) - 420x}{x(x + 20)} = \frac{12}{5}\]
\[\frac{420x + 8400 - 420x}{x^{2} + 20x} = \frac{12}{5}\]
\[5 \cdot 8400 = 12x^{2} + 240x\]
\[12x² + 240x - 42\ 000 = 0\ \ \ \ \ |\ :12\]
\[x^{2} + 20x - 3500 = 0\]
\[D = b^{2} - 4ac = 400 - 4 \cdot 1 \cdot ( - 3500) =\]
\[= 14\ 400\]
\[x_{1} = \frac{- 20 + 120}{2} = \frac{100}{2} =\]
\[= 50\ \left( \frac{км}{ч} \right) - скорость\ ІІ\ авто.\]
\[x_{2} = \frac{- 20 - 120}{2} = - \frac{140}{2} = - 70 < 0 \Longrightarrow не\ \]
\[подходит.\]
\[Ответ:первый\ автомобиль\ ехал\ со\ \]
\[скоростью\ 70\ \frac{км}{ч}.\]
\[\frac{18}{x} - \frac{46}{x - 5} = 1\]
\[\frac{18 \cdot (x - 5) - 45x}{x(x - 5)} = x(x - 5);\ \ \ \ \ x \neq 0;\ \ \]
\[x \neq 5\]
\[18x - 90 - 46x = x^{2} - 5x\]
\[x^{2} - 5x + 28x + 90 = 0\]
\[x^{2} + 23x + 90 = 0\]
\[x_{1} + x_{2} = - 23\]
\[x_{1} \cdot x_{2} = 90 \Longrightarrow x_{1} = - 18\ \ \ и\ \ \ x_{2} = - 5\]
\[Ответ:\ \ x = - 18\ \ и\ \ \ x = - 5.\]
\[\left\{ \begin{matrix} 7x - y = 5 \\ \text{xy} = 18\ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\left\{ \begin{matrix} y = 7x - 5\ \ \ \ \ \ \ \ \\ x(7x - 5) = 18 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]
\[\left\{ \begin{matrix} y = 7x - 5\ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 7x² - 5x - 18 = 0 \\ \end{matrix} \right.\ \]
\[7x^{2} - 5x - 18 = 0\]
\[D = b^{2} - 4ac = 25 - 4 \cdot 7 \cdot ( - 18) =\]
\[= 25 + 504 = 529\]
\[x_{1} = \frac{5 + 23}{14} = \frac{28}{14} = 2\]
\[x_{2} = \frac{5 - 23}{14} = - \frac{18}{14} = - \frac{9}{7} = - 1\frac{2}{7}\]
\[\left\{ \begin{matrix} y = 7x - 5 \\ \left\lbrack \begin{matrix} x_{1} = 2\ \ \ \ \ \ \ \\ x_{2} = - 1\frac{2}{7} \\ \end{matrix} \right.\ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ }\left\lbrack \begin{matrix} \left\{ \begin{matrix} y = 7 \cdot 2 - 5 \\ x = 2\ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ } \\ \left\{ \begin{matrix} y = 7 \cdot \left( - \frac{9}{7} \right) - 5\ \ \\ x = - 1\frac{2}{7}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \end{matrix} \right.\ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\]
\[\left\lbrack \begin{matrix} \left\{ \begin{matrix} y = 9 \\ x = 2 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ } \\ \left\{ \begin{matrix} y = - 14 \\ x = - 1\frac{2}{7} \\ \end{matrix} \right.\ \\ \end{matrix} \right.\ \]
\[Ответ:\ \ (2;9);\ \ \left( - 1\frac{2}{7};\ - 14 \right)\text{.\ }\]