Задача 10. Решите систему уравнений \(\left\{ {\begin{array}{*{20}{c}} {\sqrt[x]{{x + 2y}} = 2\,\,\,\,\,\,\,\,\,\,\,\,} \\ {\left( {2x + 4y} \right)\,{3^x} = 72} \end{array}} \right.\)
Ответ
ОТВЕТ: \(\left( {2;\,1} \right).\)
Решение
\(\left\{ {\begin{array}{*{20}{c}}{\sqrt[x]{{x + 2y}} = 2,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\left( {2x + 4y} \right) \cdot {3^x} = 72}\end{array}} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x + 2y = {2^x},\,\,\,\,\,\,\,\,\,\,\,\,}\\{\left( {x + 2y} \right) \cdot {3^x} = 36}\end{array}} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x + 2y = {2^x},}\\{{2^x} \cdot {3^x} = 36\,\,\,}\end{array}} \right.\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x + 2y = {2^x},}\\{{6^x} = {6^2}\,\,\,\,\,\,\,\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 2,\\y = 1.\end{array} \right.\)
Ответ: \(\left( {2;1} \right).\)