Solve the system of Linear equations
$x + 2 y + z = 8$
$2 x + y - z = 1$
$x - y + z = 2$

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Solution

The system of equations can be written in the matrix form as
$⎛ ⎜ ⎝ \begin{matrix} 1 & 2 & 1 2 & 1 & - 1 1 & - 1 & 1 \end{matrix} ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ \begin{matrix} x y z \end{matrix} ⎞ ⎟ ⎠ = ⎛ ⎜ ⎝ \begin{matrix} 812 \end{matrix} ⎞ ⎟ ⎠$
$⎛ ⎜ ⎝ \begin{matrix} 1 & 2 & 1 2 & 1 & - 1 1 & - 1 & 1 \end{matrix} ⎞ ⎟ ⎠$ should be reduced.
$⎛ ⎜ ⎝ \begin{matrix} 1 & 2 & 1 2 & 1 & - 1 1 & - 1 & 1 \end{matrix} ⎞ ⎟ ⎠ R 2 - 2 R 1, R 3 - R 1 - ----------- \to ⎛ ⎜ ⎝ \begin{matrix} 1 & 2 & 1 0 & - 3 & - 3 0 & - 3 & 0 \end{matrix} ⎞ ⎟ ⎠ R 3 - R 2 - ----- \to ⎛ ⎜ ⎝ \begin{matrix} 1 & 2 & 1 0 & - 3 & - 3 0 & 0 & 3 \end{matrix} ⎞ ⎟ ⎠$
$\Rightarrow ⎛ ⎜ ⎝ \begin{matrix} 1 & 2 & 1 0 & - 3 & - 3 0 & 0 & 3 \end{matrix} ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ \begin{matrix} x y z \end{matrix} ⎞ ⎟ ⎠ = ⎛ ⎜ ⎝ \begin{matrix} 812 \end{matrix} ⎞ ⎟ ⎠$

$\Rightarrow x + 2 y + z = 8, - 3 y - 3 z = 1, 3 z = 2$
$\Rightarrow z = \frac{2}{3}, y = - 1, x = \frac{28}{3}$

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