Решите систему уравнений: x-3y-2z=-2; -x+y+5z=13; 3x+y-4z=-10.

\[\left\{ \begin{matrix} x - 3y - 2z = - 2\ \ ( + ) \\ - x + y + 5z = 13\ \ ( - ) \\ 3x + y - 4z = - 10\ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ }\]

\(\overline{( + ) - 2y + 3z = 11\ \ \ \ \ \ }\)

\(( - ) - 4x + 9z = 23\)

\[\frac{9 \cdot (6x + 9)}{5} - 4x = 23\ \ \ \ | \cdot 5\]

\[54x + 81 - 20x = 115\]

\[34x = 34\]

\[x = 1\ \]

\[\left\{ \begin{matrix} 3z - 2y = 11^{*}\text{\ \ \ \ \ \ } \\ 9z - 4x = 23^{**}\text{\ \ \ \ \ } \\ y = 4z - 3x - 10 \\ \end{matrix} \right.\ \]

\[*\ \ 3z - 2 \cdot (4z - 3x - 10) = 11\]

\[\ \ \ 3z - 8z + 6x + 20 = 11\]

\[\ \ \ 5z = 6x + 9\]

\[\ \ \ z = \frac{6x + 9}{5}\]

\[\left\{ \begin{matrix} x = 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 9z - 4 \cdot 1 = 23 \\ 3z - 2y = 11\ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 9z = 27\ \ \ \ \ \ \ \ \ \ \ \\ 3z - 2y = 11 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} x = 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ z = 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 3 \cdot 3 - 2y = 11 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = 1\ \ \ \ \ \ \ \ \ \ \ \ \ \\ z = 3\ \ \ \ \ \ \ \ \ \ \ \ \ \\ 2y = 9 - 11 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = 1\ \ \ \ \ \ \\ z = 3\ \ \ \ \ \ \\ 2y = - 2 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = 1\ \ \ \\ y = - 1 \\ z = 3\ \ \ \\ \end{matrix} \right.\ \]

\[Ответ:(1;\ - 1;3)\text{.\ }\]