Common solution to matrix pair system of equations

Question

Consider $Ax = b$ and $Bx = c$. Assuming that we know both A and B are rank 2 and have 3 linearly independent solutions, how do we go about showing that they share a common solution?

And that the solution is unique?

Answer 1

If you place the system as follows: $$\begin{bmatrix}A\\B\end{bmatrix}x=\begin{bmatrix}b\\c\end{bmatrix}$$

The solutions to this system are the common solutions to both original equations. You may now see that if there's a solution, it won't necessarily be unique, because if $A=B$ and $b=c$ you will have that both matrix have rank $2$ and have three linearly independent solutions, therefore you will have three linearly independent solutions common to both systems.

Also, you can easily check that you don't always have a common solution. For example, take $b\neq0$, $c=-b$ and $A=B$. Then both matrix have rank $2$ and there's no common solution, because as $b\neq0$, then $c\neq b$, so if $x$ solves $Ax=b$: $$\begin{bmatrix}A\\B\end{bmatrix}x=\begin{bmatrix}b\\b\end{bmatrix}\neq\begin{bmatrix}b\\c\end{bmatrix}$$ So there isn't any common solution to both systems.