Системы иррациональных уравнений. Задача 7

ОТВЕТ: \(\left( {2;7} \right),\;\left( {\dfrac{5}{4};1} \right),\;\left( {5;-5} \right)\).

\(\left\{ {\begin{array}{*{20}{c}}{\sqrt[3]{{\dfrac{{y + 1}}{{x-1}}}}-2\sqrt[3]{{\dfrac{{x-1}}{{y + 1}}}} = 1,\,\,\,\,\,\,}\\{\sqrt {x + y}  + \sqrt {x-y + 6}  = 4.}\end{array}} \right.\)

Рассмотрим первое уравнение. Пусть \(\sqrt[3]{{\dfrac{{y + 1}}{{x-1}}}} = t\). Тогда:

\(t-\dfrac{2}{t} = 1\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}{t^2}-t-2 = 0,\\t \ne 0\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left[ \begin{array}{l}t = -1,\\t = 2.\end{array} \right.\)

Возвращаясь к прежним переменным получим:

\(\left[ \begin{array}{l}\sqrt[3]{{\dfrac{{y + 1}}{{x-1}}}} = 2,\\\sqrt[3]{{\dfrac{{y + 1}}{{x-1}}}} = -1\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left[ \begin{array}{l}y = 8x-9,\\y = -x.\end{array} \right.\)

Тогда исходная система равносильна совокупности двух систем уравнений:

\(\left[ \begin{array}{l}\left\{ {\begin{array}{*{20}{c}}{y = 8x-9,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\sqrt {x + y}  + \sqrt {x-y + 6}  = 4,}\end{array}} \right.\\\left\{ {\begin{array}{*{20}{c}}{y = -x,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\sqrt {x + y}  + \sqrt {x-y + 6}  = 4.}\end{array}} \right.\end{array} \right.\)

Рассмотрим первую систему совокупности:

\(\left\{ {\begin{array}{*{20}{c}}{y = 8x-9,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\sqrt {x + y}  + \sqrt {x-y + 6}  = 4}\end{array}} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}y = 8x-9,\\\sqrt {9x-9}  + \sqrt {15-7x}  = 4.\end{array} \right.\)

\(\sqrt {9x-9}  = 4-\sqrt {15-7x} \,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}4-\sqrt {15-7x}  \ge 0,\\9x-9 = 16-8\sqrt {15-7x}  + 15-7x\end{array} \right.\,\,\,\,\, \Leftrightarrow \)

\(\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}-\dfrac{1}{7} \le x \le \dfrac{{15}}{7},\\\sqrt {15-7x}  = 5-2x\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}-\dfrac{1}{7} \le x \le \dfrac{{15}}{7},\\5-2x \ge 0,\\4{x^2}-13x + 10 = 0\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left[ \begin{array}{l}x = 2,\\x = \dfrac{5}{4}.\end{array} \right.\)

\(\left\{ {\begin{array}{*{20}{c}}{y = 8x-9,}\\{\,\left[ \begin{array}{l}x = 2,\\x = \dfrac{5}{4}\end{array} \right.\,\,\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left[ \begin{array}{l}\left\{ \begin{array}{l}x = 2,\\y = 7,\end{array} \right.\\\left\{ \begin{array}{l}x = \dfrac{5}{4},\\y = 1.\end{array} \right.\end{array} \right.\)

Рассмотрим вторую систему совокупности:

\(\left\{ {\begin{array}{*{20}{c}}{y = -x,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\sqrt {x + y}  + \sqrt {x-y + 6}  = 4}\end{array}} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}y = -x,\\\sqrt {2x + 6}  = 4\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left[ \begin{array}{l}x = 5,\\y = -5.\end{array} \right.\)

Ответ:  \(\left( {2;7} \right),\,\,\,\,\,\left( {\dfrac{5}{4};1} \right),\,\,\,\,\,\left( {5;-5} \right).\)