Describe all solutions of the linear system system Describe all solutions of the system (the last column is the augmented column)
$-x_1 +2x_2 + x_3 + 4x_4 = 0$
$2x_1 + x_2 -x_3 +x_4 = 1$
$\pmatrix{-1&2&1&4&0 \\
          2&1&-1&1&1
}
\sim \pmatrix{1&-2&-1&-4&0 \\
              0&5&1&9&1}
\sim \pmatrix{1&0 & -\dfrac{3}{5}& -\dfrac{2}{5}& \dfrac{2}{5} \\
              0&5&1&9&1}
$
Solving we get the equations:
$x_1 = \dfrac{3}{5}x_3 + \dfrac{2}{5}x_4 + \dfrac{2}{5} \iff x_1 = 3x_3 + 2x_4 + 2$    (Can I scale like this?)
$5x_2 = -x_3 -9x_4 +1 \iff x_2 = x_3 +9x_4 - 1$ (can I scale like this?)
Now we have:
$\langle x_1, x_2, x_3, x_4\rangle = \langle 3x_3 + 2x_4 +2, -x_3 -9x_4 + 1, x_3, x_4\rangle$
$= x_3\langle 3, -1, 1, 0 \rangle + x_4\langle 2, -9, 0, 1\rangle + \langle 2,1,0,0\rangle
$
Where did I go wrong? 

You need to be careful scaling.
You have the equations:
$$-x_1+\frac{3}{5}x_3 + \frac{2}{5}x_4 = -\frac{2}{5}, \ \ \ 5 x_2 + x_3 + 9x_4  = 1$$
which can be written as
$$x_1 = \frac{3}{5}x_3 + \frac{2}{5}x_4 + \frac{2}{5}, \ \ \ x_2 = -\frac{1}{5} x_3 - \frac{9}{5} x_4 + \frac{1}{5}$$
Thus all the solutions are:
$$x = \frac{1}{5} \pmatrix{3 \\ -1 \\ 5 \\ 0} x_3 + \frac{1}{5} \pmatrix{2 \\ -9 \\ 0 \\ 5} x_4 + \frac{1}{5} \pmatrix{2 \\ 1 \\ 0 \\ 0}$$
