Найдите решение системы уравнений 3*√(x/y)+2*√(y/x)=5; 4√x+√y=10.

Ответ:

\[\left\{ \begin{matrix} 3\sqrt{\frac{x}{y}} + 2\sqrt{\frac{y}{x}} = 5 \\ 4\sqrt{x} + \sqrt{y} = 10 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\sqrt{\frac{x}{y}} = t:\]

\[3t + \frac{2}{t} = 5\]

\[3t^{2} + 2 - 5t = 0\]

\[3t^{2} - 5t + 2 = 0\]

\[D = 25 - 24 = 1\]

\[1)\ \left\{ \begin{matrix} \sqrt{\frac{x}{y}} = 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 4\sqrt{x} + \sqrt{y} = 10 \\ \end{matrix} \right.\ \ \]

\[\left\{ \begin{matrix} \sqrt{x} = \sqrt{\text{y\ }}\text{\ \ \ \ \ \ \ \ \ \ \ \ } \\ 4\sqrt{x} + \sqrt{y} = 10 \\ \end{matrix} \right.\ \]

\[4\sqrt{y} + \sqrt{y} = 10\]

\[5\sqrt{y} = 10\]

\[\sqrt{y} = 2 \Longrightarrow y = 4.\]

\[\sqrt{x} = 2 \Longrightarrow x = 4.\]

\[2)\ \left\{ \begin{matrix} \sqrt{\frac{x}{y}} = \frac{2}{3}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ 4\sqrt{x} + \sqrt{y} = 10 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\left\{ \begin{matrix} \sqrt{x} = \frac{2}{3}\sqrt{y}\text{\ \ \ \ \ \ \ \ \ } \\ 4\sqrt{x} + \sqrt{y} = 10 \\ \end{matrix} \right.\ \ \]

\[4 \cdot \frac{2}{3}\sqrt{y} + \sqrt{y} = 10\]

\[\frac{8}{3}\sqrt{y} + \sqrt{y} = 10\]

\[\frac{11\sqrt{y}}{3} = 10\]

\[\sqrt{y} = \frac{30}{11} \Longrightarrow y = \frac{900}{121};\]

\[\sqrt{x} = \frac{2}{3} \cdot \frac{30}{11} = \frac{20}{11} \Longrightarrow x = \frac{400}{121}.\]

\[Ответ:(4;4);\left( \frac{400}{121};\frac{900}{121} \right).\]