Answer
Consistent
Solution set: $\left\{\left(x,y,z\right)|x=2,y=z-3,z\text{ is any real number}\right\}$
Work Step by Step
We are given the system of equations:
$\begin{cases}
x-y+z=5\\
3x+2y-2z=0
\end{cases}$
Write the augmented matrix:
$\begin{bmatrix}
1&-1&1&|&5\\3&2&-2&|&0\end{bmatrix}$
Perform row operations to bring the matrix to the reduced row echelon form:
$R_2=-3r_1+r_2$
$\begin{bmatrix}
1&-1&1&|&5\\0&5&-5&|&-15\end{bmatrix}$
$R_2=\dfrac{1}{5}r_2$
$\begin{bmatrix}1&-1&1&|&5\\0&1&-1&|&-3\end{bmatrix}$
$R_1=r_2+r_1$
$\begin{bmatrix}1&0&0&|&2\\0&1&-1&|&-3\end{bmatrix}$
The system has two equations and 3 variables; therefore it has infinitely many solutions.
Write the corresponding system of equations:
$\begin{cases}
x=2\\
y-z=-3
\end{cases}$
Express $y$ in terms of $z$:
$x=2$
$y-z=-3\Rightarrow y=z-3$
The system is consistent. The solution set is:
$\left\{\left(x,y,z\right)|x=2,y=z-3,z\text{ is any real number}\right\}$
