Answer
$(-\frac{3}{5}+\frac{2}{5}t, \frac{1}{5}+\frac{1}{5}t, t)$
Work Step by Step
Step 1 Write the augmented matrix of the system of equation:
$\begin{bmatrix} 1&3&-1&0\\3&4&-2&-1\\-1&2&0&1\end{bmatrix}
\begin{array} \\ \\3R_1-R_2\to R_2 \\R_1+R_3\to R_3 \end{array}$
Step 2. Perform the row operations shown on the right side of the matrix:
$\begin{bmatrix} 1&3&-1&0\\0&5&-1&1\\0&5&-1&1\end{bmatrix}
\begin{array} \\ \\ \\R_3-R_2\to R_3 \end{array}$
Step 3. Perform the row operations shown on the right side of the matrix:
$\begin{bmatrix} 1&3&-1&0\\0&5&-1&1\\0&0&0&0\end{bmatrix}
\begin{array} \\ \\ \\R_3-R_2\to R_3 \end{array}$
Step 4. Perform the row operations shown on the right side of the matrix:
$\begin{bmatrix} 1&3&-1&0\\0&5&-1&1\\0&0&0&0\end{bmatrix}
\begin{array} \\ \\R_2/5 \to R_2 \\ \\ \end{array}$
Step 5. Perform the row operations shown on the right side of the matrix:
$\begin{bmatrix} 1&3&-1&0\\0&1&-1/5&1/5\\0&0&0&0\end{bmatrix}
\begin{array} \\R_1-3R_2\to R_1 \\ \\ \\ \end{array}$
Step 5. Perform the row operations shown on the right side of the matrix:
$\begin{bmatrix} 1&0&-2/5&-3/5\\0&1&-1/5&1/5\\0&0&0&0\end{bmatrix}$
Step 6. Let $z=t$, the middle equation gives $y=\frac{1}{5}+\frac{1}{5}t$ and the first equation gives $x=-\frac{3}{5}+\frac{2}{5}t$
Step 7. Conclusion: the answers to the system are $(-\frac{3}{5}+\frac{2}{5}t, \frac{1}{5}+\frac{1}{5}t, t)$
