Finding the vector form of the general solution Ax = 0 
Suppose that $x_1 = -1$, $x_2 = 2$, $x_3 = 4$, $x_4 = -3$ is a solution of a non-homogeneous linear system $A\mathbf{x} = \mathbf{b}$ and that the solution set of the homogeneous system $A\mathbf{x} =\mathbf{0}$ is given by the formulas: 
  $$\begin{align*}
x_1 &= -3r + 4s,\\
x_2 &= r - s,\\
x_3 &= r,\\
x_4 &= s.
\end{align*}$$
  Find the vector form of the general solutions of Ax = 0 and Ax = b

I ended up with something like: 
( -3 4)
( 1 -1)
( 1 0)
( 0 1) 

where I separated the $r$ and $s$ values, I haven't tried to actually solve though because I'm kinda confused about what I'm suppose to do with this. 

The key is to find a single solution to the non-homogeneous, which can be done by Gaussian elimination if such a solution exists. The general solution is then this solution plus any linear combination of solutions to the homogeneous equation.
