Question

Решите пожалуйста 20.51

Answer 1

$\left \{ {{\frac{3}{2x+y}+\frac{1}{2x-y}=\frac{2}{5}} \atop {\frac{7}{2x+y}+\frac{2}{2x-y}=\frac{3}{5}}} \right.$

Пусть $2x+y=a, 2x-y=b, a\neq 0,b\neq 0$

$\left \{ {{\frac{3}{a}+\frac{1}{b}=\frac{2}{5}} \atop {\frac{7}{a}+\frac{2}{b}=\frac{3}{5}}} \right. \\ \\ \left \{ {{-\frac{6}{a}-\frac{2}{b}=-\frac{4}{5}} \atop {\frac{7}{a}+\frac{2}{b}=\frac{3}{5}}} \right. \\ \\ \left \{ {{\frac{1}{a}=-\frac{1}{5}} \atop {7\cdot(-\frac{1}{5})+\frac{2}{b}=\frac{3}{5}}} \right. \\ \\ \left \{ {{a=-5} \atop {-\frac{7}{5}+\frac{2}{b}=\frac{3}{5}}} \right. \\ \\ \left \{ {{a=-5} \atop {\frac{2}{b}=\frac{10}{5}}} \right.$

$\left \{ {{a=-5} \atop {\frac{2}{b}=2}} \right.\\ \\ \left \{ {{a=-5} \atop {b=1}} \right.$

Возвращаемся к исходным переменным:

$\left \{ {{2x+y=-5} \atop {2x-y=1}} \right. \\ \\ \left \{ {{4x=-4} \atop {2y=-6}} \right. \\ \\ \left \{ {{x=-1} \atop {y=-3}} \right.$

Ответ: $(-1;-3)$