Question

Решите систему уравнений

1)6/x-y - 8/x+y = -2 9/x-y + 10/x+y=8


2)9/x+y + 2/x-y =3 18/x+y - 5/x-y= -3

Answer 1

Ответ:

1)  Решить систему уравнений .

$\bf \left\{\begin{array}{l}\bf \dfrac{6}{x-y}-\dfrac{8}{x+y}=-2\\\bf \dfrac{9}{x-y}+\dfrac{10}{x+y}=8\end{array}\right$             Замена :  $\bf u=\dfrac{1}{x-y}\ \ ,\ \ v=\dfrac{1}{x+y}$   .

$\left\{\begin{array}{l}\bf 6u-8v=-2\\\bf 9u+10v=8\end{array}\right\ \ \left\{\begin{array}{l}\bf 3u-4v=-1\ \Big|\cdot (-3)\\\bf 9u+10v=8\end{array}\right\ \oplus \ \left\{\begin{array}{l}\bf 3u-4v=-1\\\bf 22v=11\end{array}\right\\\\\\\left\{\begin{array}{l}\bf 3u=4v-1\\\bf v=\dfrac{1\1}{2}\end{array}\right\ \ \left\{\begin{array}{l}\bf 3u=1\\\bf v=\dfrac{1}{2}\end{array}\right\ \ \left\{\begin{array}{l}\bf u=\dfrac{1}{3}\\\bf \ v=\dfrac{1}{2}\end{array}\righ$  

Переходим к старой переменной .

$\left\{\begin{array}{l}\bf \dfrac{1}{x-y}=\dfrac{1}{3}\\\bf \dfrac{1}{x+y}=\dfrac{1}{2}\end{array}\right\ \ \left\{\begin{array}{l}\ \ \bf x-y=3\\\\\bf x+y=2\end{array}\right\ \oplus \ \ominus \ \left\{\begin{array}{l}\bf 2x=5\\\\\bf -2y=1\end{array}\right\ \left\{\begin{array}{l}\ \ \bf x=\dfrac{5}{2}\\\bf y=-\dfrac{1}{2}\end{array}\right\\\\\\\left\{\begin{array}{l}\ \ \bf x=2,5\\\bf y=-0,5\end{array}\right\ \ \ \ \Rightarrow \ \ \ \ \bf Otvet:\ (\ 2,5\ ;-0,5\ )\ .$      

2)  Решить систему уравнений .

$\bf \left\{\begin{array}{l}\bf \dfrac{9}{x+y}+\dfrac{2}{x-y}=3\\\bf \dfrac{18}{x+y}-\dfrac{5}{x-y}=-3\end{array}\right$             Замена :  $\bf u=\dfrac{1}{x+y}\ \ ,\ \ v=\dfrac{1}{x-y}$   .

$\left\{\begin{array}{l}\bf 9u+2v=3\ \Big|\cdot (-2)\\\bf 18u-5v=-3\end{array}\right\ \oplus \ \left\{\begin{array}{l}\bf 9u+2v=3\\\bf -9v=-9\end{array}\right\ \ \left\{\begin{array}{l}\bf 9u=3-2v\\\bf v=1\end{array}\right\\\\\\\left\{\begin{array}{l}\bf 9u=1\\\bf v=1\end{array}\right\ \ \left\{\begin{array}{l}\bf u=\dfrac{1}{9}\\\bf \ v=1\end{array}\righ$  

Переходим к старой переменной .

$\left\{\begin{array}{l}\bf \dfrac{1}{x+y}=\dfrac{1}{9}\\\bf \dfrac{1}{x-y}=1\end{array}\right\ \ \left\{\begin{array}{l}\ \ \bf x+y=9\\\\\bf x-y=1\end{array}\right\ \oplus \ \ominus \ \left\{\begin{array}{l}\bf 2x=10\\\\\bf 2y=8\end{array}\right\ \left\{\begin{array}{l}\bf x=5\\\\\bf y=4\end{array}\right\\\\\\\bf Otvet:\ (\, 5\, ;\, 4\, )\ .$