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

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

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

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

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

\[\left\{ \begin{matrix}
a = 2b + 1\ \ \ \ \ \ \ \\
4b + 2 - b = 5 \\
\end{matrix} \right.\ \]

\[\left\{ \begin{matrix}
a = 2b + 1 \\
3b = 3\ \ \ \ \ \ \ \ \\
\end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix}
b = 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \\
a = 2 \cdot 1 + 1 \\
\end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix}
b = 1 \\
a = 3 \\
\end{matrix} \right.\ \]

\[Подставим:\]

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

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

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

\[Ответ:\left( \frac{1}{3};\frac{1}{2} \right).\]