Answer
Consistent
Solution set: $\{(x,y,z)|x=5z-2,y=4z-3,z\text{ is any real number}\}$
Work Step by Step
We are given the system of equations:
$\begin{cases}
-x+y+z=-1\\
-x+2y-3z=-4\\
3x-2y-7z=0
\end{cases}$
Write the augmented matrix:
$\begin{bmatrix}
-1&1&1&|&-1\\-1&2&-3&|&-4\\3&-2&-7&|&0\end{bmatrix}$
Perform row operations to bring the matrix to the reduced row echelon form:
$R_1=-r_1$
$\begin{bmatrix}
1&-1&-1&|&1\\-1&2&-3&|&-4\\3&-2&-7&|&0\end{bmatrix}$
$R_2=r_1+r_2$
$\begin{bmatrix}1&-1&-1&|&1\\0&1&-4&|&-3\\3&-2&-7&|&0\end{bmatrix}$
$R_3=-3r_1+r_3$
$\begin{bmatrix}1&-1&-1&|&1\\0&1&-4&|&-3\\0&1&-4&|&-3\end{bmatrix}$
$R_3=-r_2+r_3$
$\begin{bmatrix}1&-1&-1&|&1\\0&1&-4&|&-3\\0&0&0&|&0\end{bmatrix}$
As the last row contains only zeros, the system is consistent, with infinitely many solutions.
Write the corresponding system:
$\begin{cases}
x-y-z=1
y-4z=-3
\end{cases}$
Express $x,y$ in terms of $z$:
$y-4z=-3\Rightarrow y=4z-3$
$x-y-z=1\Rightarrow x=4z-3+z+1=5z-2$
The solutions set is:
$\{(x,y,z)|x=5z-2,y=4z-3,z\text{ is any real number}\}$
