Решите систему уравнений: y/x-x/y=16/15; 4y-5x=15.

Ответ:

\[\left\{ \begin{matrix} \frac{y}{x} - \frac{x}{y} = \frac{16}{15}\text{\ \ \ \ \ } \\ 4y - 5x = 15 \\ \end{matrix}\text{\ \ } \right.\ \]

\[Пусть\ \ \ \frac{x}{y} = t:\ \]

\[\left\{ \begin{matrix} \frac{1}{t} - t = \frac{16}{15}\ \ \ \ \ \ | \cdot 15t \\ 4y - 5x = 15\ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[15 - 15t^{2} - 16t = 0\]

\[15t^{2} + 16t - 15 = 0\]

\[D = 256 + 900 = 1156\]

\[t = \frac{- 16 + 34}{30} = \frac{9}{15};\ \ \ \ \ \ \ \ \ \ \]

\[\ t = \frac{- 16 - 34}{30} = - \frac{5}{3}.\]

\[1.\ \left\{ \begin{matrix} \frac{x}{y} = \frac{9}{15}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ } \\ 4y - 5x = 15 \\ \end{matrix}\text{\ \ \ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} x = \frac{9y}{15}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ } \\ 4y - 3y = 15 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = \frac{9y}{15} \\ y = 15 \\ \end{matrix}\text{\ \ \ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} x = 9\ \ \\ y = 15 \\ \end{matrix} \right.\ \]

\[2.\ \left\{ \begin{matrix} \frac{x}{y} = - \frac{5}{3}\text{\ \ \ \ \ \ \ \ \ \ \ } \\ 4y - 5x = 15 \\ \end{matrix}\text{\ \ \ \ \ \ } \right.\ \text{\ \ }\]

\[\left\{ \begin{matrix} x = - \frac{5y}{3}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ 4y + \frac{25y}{3} = 15\ \ \ \ \ | \cdot 3 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\left\{ \begin{matrix} 12y + 25y = 45 \\ x = - \frac{5y}{3}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \end{matrix}\ \right.\ \]

\[\left\{ \begin{matrix} 37y = 45 \\ x = - \frac{5y}{3}\ \\ \end{matrix}\text{\ \ \ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} y = \frac{45}{37}\text{\ \ \ \ } \\ x = - \frac{75}{37} \\ \end{matrix} \right.\ \]

\[Ответ:(9;15);\ \left( - 2\frac{1}{37};1\frac{8}{37} \right).\]