Answer
$(\frac{2}{11}, \frac{48}{11}, -\frac{60}{11}, -\frac{40}{11})$
Work Step by Step
Step 1. Set up the augmented matrix of the system of equations:
$\begin{array} \\A=\\ \end{array}
\begin{bmatrix} 1&0&-1&1&2\\2&1&0&-2&12\\0&3&1&1&4\\1&1&-1&0&10 \end{bmatrix}
\begin{array} \\ \\R_2-2R_1\to R_2\\ \\R_4-R_1\to R_4 \\ \end{array}$
Step 2. Do the row operations indicated on the right side of the matrix:
$\begin{array} \\A=\\ \end{array}
\begin{bmatrix} 1&0&-1&1&2\\0&1&2&-4&8\\0&3&1&1&4\\0&1&0&-1&8 \end{bmatrix}
\begin{array} \\ \\R_2-R_4\to R_2\\R_3-3R_4\to R_3 \\R_4\leftrightarrow R_2 \\ \end{array}$
Step 3. Do the row operations indicated on the right side of the matrix:
$\begin{array} \\A=\\ \end{array}
\begin{bmatrix} 1&0&-1&1&2\\0&1&0&-1&8\\0&0&1&4&-20\\0&0&2&-3&0 \end{bmatrix}
\begin{array} \\ \ \\ \\ \\ \end{array}$
Step 4. Do the operation $2R_3-R_4$ to get $11w=-40$ or $w=-\frac{40}{11}$
Step 5. Use back-substitution to get the final answer as $(\frac{2}{11}, \frac{48}{11}, -\frac{60}{11}, -\frac{40}{11})$
