Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 721

\[\boxed{\mathbf{721}.}\]

\[1)\ \left\{ \begin{matrix} 5^{x + 1} \bullet 3^{y} = 75 \\ 3^{x} \bullet 5^{y - 1} = 3\ \ \\ \end{matrix} \right.\ \ (\text{x)}\]

\[5^{x + 1} \bullet 5^{y - 1} \bullet 3^{y} \bullet 3^{x} = 75 \bullet 3\]

\[5^{x + 1 + y - 1} \bullet 3^{y + x} = 225\]

\[5^{x + y} \bullet 3^{y + x} = 225\]

\[(5 \bullet 3)^{x + y} = 225\]

\[15^{x + y} = 15^{2}\]

\[x + y = 2\]

\[y = 2 - x.\]

\[2)\ 3^{x} \bullet 5^{2 - x - 1} = 3\]

\[3^{x} \bullet 5^{1 - x} = 3\]

\[3^{x} \bullet \frac{5}{5^{x}} = 3\]

\[\frac{3^{x}}{5^{x}} = \frac{3}{5}\]

\[\left( \frac{3}{5} \right)^{x} = \left( \frac{3}{5} \right)^{1}\]

\[x = 1\ \]

\[y = 2 - 1 = 1.\]

\[Ответ:\ \ (1;1).\]

\[2)\ \left\{ \begin{matrix} 3^{x} \bullet 2^{y} = 4 \\ 3^{y} \bullet 2^{x} = 9 \\ \end{matrix} \right.\ \ (\text{x)}\]

\[3^{x} \bullet 3^{y} \bullet 2^{y} \bullet 2^{x} = 4 \bullet 9\]

\[3^{x + y} \bullet 2^{y + x} = 36\]

\[(3 \bullet 2)^{x + y} = 36\]

\[6^{x + y} = 6^{2}\]

\[x + y = 2\]

\[y = 2 - x.\]

\[1)\ 3^{x} \bullet 2^{2 - x} = 4\]

\[3^{x} \bullet \frac{2^{2}}{2^{x}} = 4\]

\[\frac{3^{x}}{2^{x}} \bullet 4 = 4\]

\[\left( \frac{3}{2} \right)^{x} = 1\]

\[\left( \frac{3}{2} \right)^{x} = \left( \frac{3}{2} \right)^{0}\ \]

\[x = 0;\ \]

\[y = 2 - 0 = 2.\]

\[Ответ:\ \ (0;2).\]