Solution
= Solution
A particular solution is $x_p=(1,0,-1,0)^T$, as direct substitution into all five equations verifies. Add any vector in the <null space> to obtain every solution:
$$
\boxed{x=\begin{pmatrix}1\\0\\-1\\0\end{pmatrix}+t\begin{pmatrix}-1\\1\\-1\\1\end{pmatrix},\qquad t\in\mathbb R.}
$$
Conversely, the difference of any two solutions is annihilated by $A$, so there are no further solutions. This is the <affine solution space> of a consistent <linear system>.