Решите систему уравнений: 1/x+1/y=1; 1/2x+2/y=8.

\[\left\{ \begin{matrix}
\frac{1}{x} + \frac{1}{y} = 1\ \ \\
\frac{1}{2x} + \frac{2}{y} = 8 \\
\end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

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

\[\left\{ \begin{matrix}
2a + b = 1\ \ | \cdot ( - 2) \\
a + 2b = 8\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\
\end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix}
- 4a - 2b = - 2\ \ (1) \\
a + 2b = 8\ \ \ \ \ \ \ \ \ \ \ (2) \\
\end{matrix} \right.\ \]

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

\[- 3a = 6\]

\[a = - 2.\]

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

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

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

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

\[Подставим:\ \]

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

\[\left\{ \begin{matrix}
1 = - 4x \\
1 = 5y\ \ \ \ \\
\end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix}
x = - \frac{1}{4} \\
y = \frac{1}{5}\text{\ \ \ } \\
\end{matrix} \right.\ \]

\[Ответ:\left( - \frac{1}{4};\frac{1}{5} \right).\]