We have a $3\times 6$ matrix $A$ with rank $3$ (this is all the information we have, no matrix given). Here comes the questions:
What is the number of free variables in the solution to the system $Ax = 0$? (I know, it's 3.)
For a given $b$, are we guaranteed to have a solution to $Ax = b$?
If we have a solution, what is the dimension of the solution space?
(The asker wants us to explain without using an example.)
The matrix represents a linear application $\Bbb R^6\to\Bbb R^3$. Since the rank is $3$, the dimension of the image is $3$; then, the application is surjective, and the systen $Ax=b$ has always solution.
Now, the dimension of the kernel is $6-3=3$, so the dimension of a space of solutions is also $3$.