Решите систему уравнений: 2/x-1/y=5; 2/x+1/y=7.

Ответ:

\[\left\{ \begin{matrix} \frac{2}{x} - \frac{1}{y} = 5 \\ \frac{2}{x} + \frac{1}{y} = 7 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[Пусть\ \ \frac{2}{x} = a;\ \ \frac{1}{y} = b:\]

\[\left\{ \begin{matrix} a - b = 5\ \ (1) \\ a + b = 7\ \ (2) \\ \end{matrix} \right.\ \]

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

\[2a = 12\ \ \]

\[a = 6.\]

\[\left\{ \begin{matrix} a = 6\ \ \ \ \ \ \ \ \\ a + b = 7 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} a = 6\ \ \ \ \ \ \ \ \\ b = 7 - a \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\left\{ \begin{matrix} a = 6\ \ \ \ \ \ \ \ \\ b = 7 - 6 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\left\{ \begin{matrix} a = 6 \\ b = 1 \\ \end{matrix} \right.\ \]

\[Подставим:\]

\[\left\{ \begin{matrix} \frac{2}{x} = 6\ \ \ | \cdot x \\ \frac{1}{y} = 1\ \ \ | \cdot y \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} 2 = 6x \\ 1 = y\ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = \frac{2}{6} = \frac{1}{3} \\ y = 1\ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[Ответ:\ \ \left( \frac{1}{3};1 \right).\]