Answer
Inconsistent (no solution)
Work Step by Step
We are given the system of equations:
$\begin{cases}
2x-3y-z=0\\
-x+2y+z=5\\
3x-4y-z=1
\end{cases}$
Write the augmented matrix:
$\begin{bmatrix}
2&-3&-1&|&0\\-1&2&1&|&5\\3&-4&-1&|&1\end{bmatrix}$
Perform row operations to bring the matrix to the reduced row echelon form:
$R_1=r_2+r_1$
$\begin{bmatrix}
1&-1&0&|&5\\-1&2&1&|&5\\3&-4&-1&|&1\end{bmatrix}$
$R_2=r_1+r_2$
$\begin{bmatrix}1&-1&0&|&5\\0&1&1&|&10\\3&-4&-1&|&1\end{bmatrix}$
$R_3=-3r_1+r_3$
$\begin{bmatrix}1&-1&0&|&5\\0&1&1&|&10\\0&-1&-1&|&-14\end{bmatrix}$
$R_1=r_2+r_1$
$\begin{bmatrix}1&0&1&|&15\\0&1&1&|&10\\0&-1&-1&|&-14\end{bmatrix}$
$R_3=r_2+r_3$
$\begin{bmatrix}1&0&1&|&15\\0&1&1&|&10\\0&0&0&|&-4\end{bmatrix}$
As the last row contains only zeros to the left of the vertical bar and a nonzero element to its right, the system is inconsistent, and it has no solution.
