Answer
The system is inconsistent
Work Step by Step
The augmented matrix of the system is:
$\begin{bmatrix}
3 &1 &5 |2\\
1 &1 &-1| 1\\
2& 1 & 2 |3
\end{bmatrix}$
with reduced row-echelon form:
$\begin{bmatrix}
3 &1 &5 |2\\
1 &1 &-1| 1\\
2& 1 & 2 |3
\end{bmatrix} \approx^1 \begin{bmatrix}
1 &1 &-1| 1\\
3 &1 &5 |2\\
2& 1 & 2 |3
\end{bmatrix} \approx^2 \begin{bmatrix}
1 &1 &-1| 1\\
0 &-2 &8 |-1\\
0& -1 & 4|3
\end{bmatrix} \approx^3 \begin{bmatrix}
1 &1 &-1| 1\\
0 &1 &-4 |\frac{1}{2}\\
0& -1 & 4|3
\end{bmatrix} \approx^4 \begin{bmatrix}
1 &1 &-1| 1\\
0 &1 &-4 |\frac{1}{2}\\
0& 0 & 0|\frac{3}{2}
\end{bmatrix}$
$1.P_{12}$
$2.A_{12}(-3), A_{13}(-2)$
$3.M_2(-\frac{1}{2})$
$4.A_{23}(1)$
This system of equations is inconsistent because $rank (A) \lt rank (A^\ne)$
