Решите систему уравнений: x+y-z=5; x-y+z=5; x-y-z=3.

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

\[\text{\ \ \ \ \ \ }\overline{\ \ \ \ \ \ \ 2z = 2\ \ \ \ \ \ |\ :2}\]

\[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ z = 1\]

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

\[( - )\ 4y - 2z = 2\ \ \ \ |\ :2\]

\[( + )\ 4x - 2z = 18\ \ |\ :2\]

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

\[\left\{ \begin{matrix} z = 1\ \ \ \ \ \\ 2y = 2\ \ \\ 2x = 10 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} z = 1 \\ y = 1 \\ x = 5 \\ \end{matrix} \right.\ \Longrightarrow (5;1;1)\]

\[Ответ:\ \ (5;1;1).\]