Answer
Inconsistent (no solution)
Work Step by Step
We are given the system of equations:
$\begin{cases}
2x-2y+3z=6\\
4x-3y+2z=0\\
-2x+3y-7z=1
\end{cases}$
Write the augmented matrix:
$\begin{bmatrix}
2&-2&3&|&6\\4&-3&2&|&0\\-2&3&-7&|&1\end{bmatrix}$
Perform row operations to bring the matrix to the reduced row echelon form:
$R_3=r_1+r_3$
$\begin{bmatrix}
2&-2&3&|&6\\4&-3&2&|&0\\0&1&-4&|&7\end{bmatrix}$
$R_2=-2r_1+r_2$
$\begin{bmatrix}2&-2&3&|&6\\0&1&-4&|&-12\\0&1&-4&|&7\end{bmatrix}$
$R_1=\dfrac{1}{2}r_1$
$\begin{bmatrix}1&-1&\frac{3}{2}&|&3\\0&1&-4&|&-12\\0&1&-4&|&7\end{bmatrix}$
$R_3=-r_2+r_3$
$\begin{bmatrix}1&-1&\frac{3}{2}&|&3\\0&1&-4&|&-12\\0&0&0&|&19\end{bmatrix}$
As the last row contains only zeros at the left of the vertical bar and a nonzero element at its right side, the system is inconsistent; therefore it has no solution.