\(\left\{ {\begin{array}{*{20}{c}}{{3^{2x}}-{2^y} = 725,}\\{{3^x}-{2^{\frac{y}{2}}} = 25\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{{{\left( {{3^x}} \right)}^2}-{{\left( {{2^{\frac{y}{2}}}} \right)}^2} = 725,}\\{{3^x}-{2^{\frac{y}{2}}} = 25.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\end{array}} \right.\)
Пусть \({3^x} = a,\,\,\,{2^{\frac{y}{2}}} = b\), где \(a > 0,\,\,\,b > 0.\) Тогда:
\(\left\{ {\begin{array}{*{20}{c}}{{a^2}-{b^2} = 725,}\\{a-b = 25\,\,\,\,\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{\left( {a-b} \right)\left( {a + b} \right) = 725,}\\{a-b = 25\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{25\left( {a + b} \right) = 725,}\\{a-b = 25\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{a + b = 29,}\\{a-b = 25\,\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{a = 27,}\\{b = 2.\,\,}\end{array}} \right.\)
Вернёмся к прежним переменным:
\(\left\{ {\begin{array}{*{20}{c}}{{3^x} = 27,}\\{{2^{\frac{y}{2}}} = 2\,\,\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{{3^x} = {3^3},}\\{{2^{\frac{y}{2}}} = {2^1}}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x = 3,}\\{\frac{y}{2} = 1\,\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x = 3,}\\{y = 2.}\end{array}} \right.\)
Ответ: \(\left( {3;2} \right).\)