Answer
The system of equations is inconsistent
Work Step by Step
The augmented matrix of the system is:
$\begin{bmatrix}
3& 5 &-1 |14\\
1& 2 &1| 3\\
2 & 5 & 6|2
\end{bmatrix}$
with reduced row-echelon form:
$\begin{bmatrix}
3& 5 &-1 |14\\
1& 2 &1| 3\\
2 & 5 & 6|2
\end{bmatrix} \approx^1\begin{bmatrix}
1& 2 &1 |3\\
3& 5 &-1| 14\\
2 & 5 & 6|2
\end{bmatrix} \approx^2\begin{bmatrix}
1& 2 &1 |3\\
0& -1 &-4| 5\\
0 & 1 & 4|-4
\end{bmatrix} \approx^3\begin{bmatrix}
1& 2 &1 |3\\
0& -1 &-4| 5\\
0 & 0 & 0|1
\end{bmatrix} $
The equivalent system is:
$x_1+2x_2+x_3=3$
$-x_2-4x_3=5$
$0x_1+0x_2+0x_3=1$
This system of equations is inconsistent since $rank (A) \lt rank (A^\ne)$
