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

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

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

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

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

\[2a = 12\ \ \]

\[a = 6.\]

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

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

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

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

\[Подставим:\]

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

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

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

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