Question

Решите систему уравнений

Answer 1

Решение.

$\left\{\begin{array}{l}\bf \dfrac{2}{x+y}-\dfrac{3x}{y} =-1\\\bf \dfrac{1}{x+y}+\dfrac{8x}{y}=28\end{array}\right$                      ОДЗ:  $\bf x\ne -y\ ,\ y\ne 0$  .

Замена:   $\bf t=\dfrac{1}{x+y}\ \ ,\ \ p=\dfrac{x}{y}$  .

$\left\{\begin{array}{l}\bf 2t-3p=-1\\\bf t+8p=28\ |\cdot (-2)\end{array}\right+\ \left\{\begin{array}{l}\bf -19p=-57\\\bf t+8p=28\end{array}\right\ \left\{\begin{array}{l}\bf p=3\\\bf t=28-8p\end{array}\right\ \left\{\begin{array}{l}\bf p=3\\\bf t=4\end{array}\right$

Переходим к старым переменным.

$\left\{\begin{array}{l}\ \bf \dfrac{1}{x+y}=4\\\bf \dfrac{x}{y}=3\end{array}\right\ \ \left\{\begin{array}{l}\bf x+y=\dfrac{1}{4}\\\bf x=3y\end{array}\right\ \left\{\begin{array}{l}\bf 3y+y=\dfrac{1}{4}\\\bf x=3y\end{array}\right\ \left\{\begin{array}{l}\bf 4y=\dfrac{1}{4}\\\bf x=3y\end{array}\right\ \left\{\begin{array}{l}\bf y=\dfrac{1}{16}\\\bf x=\dfrac{3}{16}\end{array}\right$  

Ответ:  $\boldsymbol{\Big(\ \dfrac{3}{16}\ ;\ \dfrac{1}{16}\ \Big)}$  .