Answer
The system is inconsistent
Work Step by Step
The augmented matrix of the system is:
$\begin{bmatrix}
1 &2& -1 &1 |1\\
2&-3 &1 &-1| 2\\
1 & -5 & 2 & -2|1\\
4 & 1 & -1 & 1 |3
\end{bmatrix}$
with reduced row-echelon form:
$\begin{bmatrix}
1 &2& -1 &1 |1\\
2&-3 &1 &-1| 2\\
1 & -5 & 2 & -2|1\\
4 & 1 & -1 & 1 |3
\end{bmatrix}\approx^1\begin{bmatrix}
1 &2& -1 &1 |1\\
0&-7 &3 &-3|0\\
0 &-7 &3 &-3|0\\
0 &-7 &3 &-3 |-1
\end{bmatrix} \approx^2 \begin{bmatrix}
1 &2& -1 &1 |1\\
0&1 &-\frac{3}{7} &\frac{3}{7}|0\\
0 &-7 &3 &-3|0\\
0 &-7 &3 &-3 |-1
\end{bmatrix} \approx^3 \begin{bmatrix}
1 &2& -1 &1 |1\\
0&1 &-\frac{3}{7} &\frac{3}{7}|0\\
0 &0 &0 &0|0\\
0 &0 &0 &0 |-1
\end{bmatrix} \approx^4 \begin{bmatrix}
1 &2& -1 &1 |1\\
0&1 &-\frac{3}{7} &\frac{3}{7}|0\\
0 &0 &0 &0|-1\\
0 &0 &0 &0 |0
\end{bmatrix} \approx^5 \begin{bmatrix}
1 &2& -1 &1 |1\\
0&1 &-\frac{3}{7} &\frac{3}{7}|0\\
0 &0 &0 &0|1\\
0 &0 &0 &0 |0
\end{bmatrix}$
The system is inconsistent since $rank(A) \lt rank (A^\ne)$
