Решите систему уравнений: 3/x+5/y=11; 8/x-7/y=9.

Ответ:

\[\left\{ \begin{matrix} \frac{3}{x} + \frac{5}{y} = 11 \\ \frac{8}{x} - \frac{7}{y} = 9\ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

\[Пусть\ \ \frac{1}{x} = a;\text{\ \ }\frac{1}{y} = b:\]

\[\left\{ \begin{matrix} 3a + 5b = 11\ \ | \cdot 7 \\ 8a - 7b = 9\ \ \ \ | \cdot 5 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} 21a + 35b = 77\ \ \ (1) \\ 40a - 35b = 45\ \ \ (2) \\ \end{matrix} \right.\ \]

\[\Longrightarrow (1) + (2):\]

\[61a = 122\ \ \]

\[a = 2.\]

\[\left\{ \begin{matrix} a = 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 3a + 5b = 11 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} a = 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 5b = 11 - 3a \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} a = 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ b = \frac{11}{5} - \frac{3}{5} \cdot 2\ \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} a = 2\ \ \ \ \ \ \ \ \ \ \ \\ b = \frac{11}{5} - \frac{6}{5} \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} a = 2\ \ \ \ \ \ \ \ \\ b = \frac{5}{5} = 1 \\ \end{matrix} \right.\ \]

\[Подставим:\ \]

\[\left\{ \begin{matrix} \frac{1}{x} = 2\ \ \ | \cdot x \\ \frac{1}{y} = 1\ \ \ | \cdot y \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} 1 = 2x \\ 1 = y\ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = \frac{1}{2} \\ y = 1 \\ \end{matrix} \right.\ \]

\[Ответ:\left( \frac{1}{2};1 \right).\]