Answer
Inconsistent (no solution)
Work Step by Step
We are given the system of equations:
$\begin{cases}
2x-2y-2z=2\\
2x+3y+z=2\\
3x+2y=0
\end{cases}$
Write the augmented matrix:
$\begin{bmatrix}
2&-2&-2&|&2\\2&3&1&|&2\\3&2&0&|&0\end{bmatrix}$
Perform row operations to bring the matrix to the reduced row echelon form:
$R_1=\dfrac{1}{2}r_1$
$\begin{bmatrix}
1&-1&-1&|&1\\2&3&1&|&2\\3&2&0&|&0\end{bmatrix}$
$R_2=-2r_1+r_2$
$\begin{bmatrix}1&-1&-1&|&1\\0&5&3&|&0\\3&2&0&|&0\end{bmatrix}$
$R_3=-3r_1+r_3$
$\begin{bmatrix}1&-1&-1&|&1\\0&5&3&|&0\\0&5&3&|&-3\end{bmatrix}$
$R_3=-r_2+r_3$
$\begin{bmatrix}1&-1&-1&|&1\\0&5&3&|&0\\0&0&0&|&-3\end{bmatrix}$
$R_2=\dfrac{1}{5}r_2$
$\begin{bmatrix}1&-1&-1&|&1\\0&1&\frac{3}{5}&|&0\\0&0&0&|&-3\end{bmatrix}$
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.
