Answer
Inconsistent (no solution)
Work Step by Step
We are given the system of equations:
$\begin{cases}
3x-2y+2z=6\\
7x-3y+2z=-1\\
2x-3y+4z=0
\end{cases}$
Write the augmented matrix:
$\begin{bmatrix}
3&-2&2&|&6\\7&-3&2&|&-1\\2&-3&4&|&0\end{bmatrix}$
Perform row operations to bring the matrix to the reduced row echelon form:
$R_1=-r_3+r_1$
$\begin{bmatrix}
1&1&-2&|&6\\7&-3&2&|&-1\\2&-3&4&|&0\end{bmatrix}$
$R_2=-7r_1+r_2$
$\begin{bmatrix}1&1&-2&|&6\\0&-10&16&|&-43\\2&-3&4&|&0\end{bmatrix}$
$R_3=-2r_3+r_2$
$\begin{bmatrix}1&1&-2&|&6\\0&-10&16&|&-43\\0&-5&8&|&-12\end{bmatrix}$
$R_3=-2r_3+r_2$
$\begin{bmatrix}1&1&-2&|&6\\0&-10&16&|&-43\\0&0&0&|&-19\end{bmatrix}$
$R_2=-\dfrac{1}{10}r_2$
$\begin{bmatrix}1&1&-2&|&6\\0&1&-1.6&|&4.3\\0&0&0&|&-19\end{bmatrix}$
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.
