Решите систему уравнений: 3/(2x+5y)-2/(3x-10y)=4; 2/(2x+5y)+3/(3x-10y)=7.

Ответ:

\[\left\{ \begin{matrix} \frac{3}{2x + 5y} - \frac{2}{3x - 10y} = 4 \\ \frac{2}{2x + 5y} + \frac{3}{3x - 10y} = 7 \\ \end{matrix}\text{\ \ \ \ } \right.\ \]

\[Пусть\ \frac{1}{2x + 5y} = t;\ \ \ \]

\[\frac{1}{3x - 10y} = c:\]

\[\left\{ \begin{matrix} 3t - 2c = 4\ \ | \cdot 3 \\ 2t + 3c = 7\ \ | \cdot 2 \\ \end{matrix}\text{\ \ \ \ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} 9t - 6c = 12 \\ 4t + 6c = 14\ \\ \end{matrix}( + ) \right.\ \text{\ \ \ \ }\]

\[\left\{ \begin{matrix} 13t = 26\ \ \ \ \ \ \ \\ c = \frac{7 - 2t}{3}\text{\ \ \ \ } \\ \end{matrix} \right.\ \text{\ \ \ }\]

\[\left\{ \begin{matrix} t = 2 \\ c = 1 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} \frac{1}{2x + 5y} = 2\ \\ \frac{1}{3x - 10y} = 1 \\ \end{matrix}\text{\ \ \ \ } \right.\ \text{\ \ \ }\]

\[\left\{ \begin{matrix} x = \frac{2}{7}\text{\ \ \ \ \ \ \ \ \ } \\ y = \frac{1 - \frac{8}{7}}{10}\ \\ \end{matrix}\text{\ \ \ \ \ \ \ \ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} x = \frac{2}{7}\text{\ \ \ \ \ \ \ } \\ y = - \frac{1}{70} \\ \end{matrix} \right.\ \]

\[Ответ:\left( \frac{2}{7};\ - \frac{1}{70} \right).\]