Решите систему уравнений: x-3y+z=6; 2x-y+3z=9; -x+4y+5z=5.

\[\left\{ \begin{matrix} x - 3y + z = 6\ \ \ \ \ \ \ \ \ \ \ \ \ \ | \cdot 2 \\ 2x - y + 3z = 9\ \ \ \ \ ( + )( - ) \\ - x + 4y + 5z = 5\ \ ( + )( - ) \\ \end{matrix} \right.\ \]

\[( + )\ 2x + 9z = 20\]

\[( - ) - 6y - 7z = - 8\]

\[\left\{ \begin{matrix} 2x + 9z = 20\ \ \ |\ :2 \\ 6y + 7z = 8\ \ \ \ \ |\ :6 \\ z = 6 - x + 3y\ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = 10 - 4,5z\ \ \ \ \\ y = \frac{4}{3} - \frac{7}{6}\text{z\ \ \ \ \ \ \ \ \ } \\ 5x - 11y = 16^{*} \\ \end{matrix} \right.\ \]

\[- 9\frac{4}{6}z = - 19\frac{2}{6}\]

\[\ - \frac{58}{6}z = - \frac{116}{6}\]

\[\left\{ \begin{matrix} z = 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x = 10 - 4,5 \cdot 2 \\ y = \frac{4}{3} - \frac{7}{6} \cdot 2\ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} z = 2\ \ \ \ \ \ \ \ \ \ \ \\ x = 9\ \ \ \ \ \ \ \ \ \ \\ y = \frac{8 - 14}{6} \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} z = 2\ \ \ \\ x = 9\ \ \ \\ y = - 1 \\ \end{matrix} \right.\ \]

\[Ответ:\ \ (9;\ - 1;\ 2).\]