<s>[INST] Solutions for linear equation system Ax = b Two solutions are known for the linear equation system $Ax = b$. These are (for example):
$$
\begin{bmatrix}
1 \\
2
\end{bmatrix},
\begin{bmatrix}
2 \\
1
\end{bmatrix}
$$
How to determine (at least) one extra solution? How to solve problems like this in general?
 [/INST] You have the following: 
$A\begin{pmatrix}
1  \\
2
\end{pmatrix} = b$ and $A\begin{pmatrix}
2  \\
1
\end{pmatrix} = b$   $\Rightarrow $ $A\begin{pmatrix}
1  \\
2
\end{pmatrix} + A\begin{pmatrix}
2  \\
1
\end{pmatrix} = 2b$  $\Rightarrow $ $A\begin{pmatrix}
3  \\
3
\end{pmatrix}=2b$ $\Rightarrow $ $A\begin{pmatrix}
3/2  \\
3/2
\end{pmatrix}=b$
</s>