Найдите решение системы: x-y-z=0; x+y-z=6; x+y+z=8.

Ответ:

\[\left\{ \begin{matrix} x - y - z = 0 \\ x + y - z = 6 \\ x + y + z = 8 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = y + z\ \ \ \ \ \ \ \ \ \\ x + x = 8\ \ \ \ \ \ \ \ \\ y - z = 6 - x \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} 2x = 8\ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y + z = x\ \ \ \ \ \ \ \ \ \\ y - z = 6 - x \\ \end{matrix} \right.\ \]

\[\ \left\{ \begin{matrix} x = 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y + z = 4\ \ \ \ \ \ \ \ \\ y - z = 6 - 4 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y + z = 4\ \ (1) \\ y - z = 2\ \ (2) \\ \end{matrix} \right.\ \]

\[2y = 6\ \]

\[y = 3\]

\[\ \left\{ \begin{matrix} x = 4\ \ \ \ \ \ \ \ \ \ \\ y = 3\ \ \ \ \ \ \ \ \ \\ y + z = 4 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

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

\[\left\{ \begin{matrix} x = 4\ \ \ \ \ \ \ \ \\ y = 3\ \ \ \ \ \ \ \\ z = 4 - 3 \\ \end{matrix} \right.\ \ \]

\[\left\{ \begin{matrix} x = 4 \\ y = 3 \\ z = 1 \\ \end{matrix} \right.\ \ \]

\[Ответ:(4;3;1).\]