Solve the following equation system using Gaussian Elimination. Solve the following equation system using Gaussian Elimination.
$x_1+2x_2-x_3-x_4+x_5=0$
$x_1+2x_2-2x_4+4x_5=0$
$2x_1+4x_2-2x_3-2x_4+2x_5=0$
$-2x_1-4x_2+4x_3+4x_5=0$
My working so far
Putting the equation system into a coefficient matrix:
$$ \left[
      \begin{array}{ccccc|c}
        1&2&-1&-1&1&0\\
        1&2&0&-2&4&0\\
        2&4&-2&-2&2&0\\
        -2&-4&4&0&4&0
      \end{array}
    \right]$$
Using Gaussian Elimination, the matrix reduces to:
$$ \left[
      \begin{array}{ccccc|c}
        1&2&-1&-1&1&0\\
        0&0&1&-1&3&0\\
        0&0&0&0&0&0\\
        0&0&0&0&0&0
      \end{array}
    \right]$$
So the equation system is:
$x_1+2x_2-x_3-x_4+x_5=0$
$x_3-x_4+3x_5=0$
This is the point I am up to. I have $2$ equations in $5$ variables, so I will need free variables/parameters. But what do I need to do to get these and have a general solution?
Thanks in advance. 
 A: Note in all combinations the last column will stay zero so we will just drop it.
$$
\begin{pmatrix}
 1 &  2 & -1 & -1 & 1 \\
 1 &  2 &  0 & -2 & 4 \\
 2 &  4 & -2 & -2 & 2 \\
-2 & -4 &  4 &  0 & 4 \\
\end{pmatrix}
\to
\begin{pmatrix}
 1 &  2 & -1 & -1 & 1 \\
 0 &  0 &  1 & -1 & 3 \\
 0 &  0 &  0 &  0 & 0 \\
 0 &  0 &  1 & -1 & 3 \\
\end{pmatrix}
\to
\begin{pmatrix}
 1 &  2 &  0 & -2 & 4 \\
 0 &  0 &  1 & -1 & 3 \\
 0 &  0 &  0 &  0 & 0 \\
 0 &  0 &  0 &  0 & 0 \\
\end{pmatrix}
$$
Which indeed leaves $x_1+2x_2-2x_4+4x_5=0$ and $x_3 - x_4 + 3x_5=0$. Write $x_1,x_3$ in terms of others, to get $x_1 = -2x_2+2x_4-4x_5$ and $x_3 = x_4 - 3x_5$. Now the general solution is
$$
\begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \end{pmatrix}
 = \begin{pmatrix} -2x_2+2x_4-4x_5 \\ x_2 \\ x_4 - 3x_5 \\ x_4 \\ x_5 \end{pmatrix}
 =
\begin{pmatrix}
-2 & 2 & -4\\
 1 & 0 &  0\\
 0 & 1 & -3\\
 0 & 1 &  0\\
 0 & 0 &  1
\end{pmatrix}
\begin{pmatrix}x_2 \\ x_4 \\ x_5 \end{pmatrix}
$$
where $x_2, x_4, x_5$ can take on any real values. In other words any linear combination of the columns of this matrix will be a solution, and no other vector:
$$
\begin{pmatrix}
-2 & 2 & -4\\
 1 & 0 &  0\\
 0 & 1 & -3\\
 0 & 1 &  0\\
 0 & 0 &  1
\end{pmatrix}
$$
