Решите систему уравнений: x/y+y/x=16/3; x-y=6.

Ответ:

\[\left\{ \begin{matrix} x = y + 6\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 3 \cdot (y + 6)^{2} + 3y^{2} = 16y(y + 6) \\ \end{matrix} \right.\ \]

\[3 \cdot \left( y^{2} + 12y + 36 \right) + 3y^{2} =\]

\[= 16y^{2} + 96y\]

\[3y^{2} + 36y + 108 + 3y^{2} =\]

\[= 16y^{2} + 96y\]

\[16y^{2} - 6y^{2} - 36y + 96y - 108 = 0\]

\[10y^{2} + 60y - 108 = 0\ \ \ \ \ |\ :2\]

\[5y^{2} + 30y - 54 = 0\]

\[D_{1} = 225 + 270 = 495\]

\[y_{1} = \frac{- 15 + 3\sqrt{55}}{5};\ \]

\[\ y_{2} = \frac{- 15 - 3\sqrt{55}}{5}.\]

\[x_{1} = \frac{- 15 + 3\sqrt{55}}{5} + 6 =\]

\[= \frac{- 15 + 3\sqrt{55} + 30}{5} =\]

\[= \frac{15 + 3\sqrt{55}}{5};\]

\[x_{2} = \frac{- 15 - 3\sqrt{55}}{5} + 6 =\]

\[= \frac{- 15 - 3\sqrt{55} + 30}{5} =\]

\[= \frac{15 - 3\sqrt{55}}{5}.\]