Решите систему уравнений: 2/(x-2y)+1/(x+2y)=7; 15/(x-2y)-2/(x+2y)=24.

Ответ:

\[\left\{ \begin{matrix} \frac{2}{x - 2y} + \frac{1}{x + 2y} = 7\ \ \\ \frac{15}{x - 2y} - \frac{2}{x + 2y} = 24 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

\[Пусть\ \ \ \frac{1}{x - 2y} = t;\ \ \ \ \ \ \]

\[\frac{1}{x + 2y} = a:\]

\[\left\{ \begin{matrix} 2t + a = 7\ \ \ \ | \cdot 2 \\ 15t - 2a = 24\ \ \ \ \\ \end{matrix}\text{\ \ \ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} 4t + 2a = 14\ \ \\ 15t - 2a = 24 \\ \end{matrix}\ ( + ) \right.\ \text{\ \ }\]

\[\left\{ \begin{matrix} 19t = 38\ \ \ \\ a = 7 - 2t \\ \end{matrix}\text{\ \ \ \ \ }\left\{ \begin{matrix} t < 2 \\ a = 3 \\ \end{matrix} \right.\ \right.\ \]

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

\[\left\{ \begin{matrix} 2x - 4y = 1\ \ \ \ | \cdot 3 \\ 3x + 6y = 1\ \ \ \ | \cdot 2 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} 6x - 12y = 3 \\ 6x + 12y = 2 \\ \end{matrix} \right.\ \ \ ( + )\]

\[\left\{ \begin{matrix} 12x = 5\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y = \frac{1 - 3x}{6}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} x = \frac{5}{12}\text{\ \ \ } \\ y = - \frac{1}{24} \\ \end{matrix} \right.\ \]

\[Ответ:\left( \frac{5}{12};\ - \frac{1}{24} \right).\]