Gauss elimination over $\mathbb{Z}_2$

Question

I should solve this system of equations over $\mathbb{Z}_2$:

$3x_1 + x_2 + 2x_3 + x_4 = 2$

$-x_1 + x_2 + x_3 - x_4 = 1$

$-5x_1 + x_2 + 3x_3 + 3x_4 = 1$

So I tried to set up the matrix, which would be

$\begin{pmatrix} 1 & 1 & 0 & 1 | 0\\ 1 & 1 & 1 & 1 | 1\\ 1 & 1 & 1 & 1 | 1 \\ \end{pmatrix}$

so line 2 and 3 are equal, then $\begin{pmatrix} 1 & 1 & 0 & 1 | 0\\ 1 & 1 & 1 & 1 | 1\\ 0 & 0 & 0 & 0 | 0 \\ \end{pmatrix}$

then subtracted the first from the second line

$\begin{pmatrix} 1 & 1 & 0 & 1 | 0\\ 0 & 0 & 1 & 0 | 1\\ 0 & 0 & 0 & 0 | 0 \\ \end{pmatrix}$

So I would be having I. $x_3 = 1$ and II. $x_1 + x_2 + x_4 = 0$, but there are many cases which would fulfill II.

Are my calculations correct up to here and how could I write the solutions for II. nicely?

Answer 1

Your work looks good. To express the multiple solutions for $x_1$, $x_2$, and $x_4$, you can allow $x_2$ and $x_4$ to be arbitrary, and then we have $$x_1=x_2+x_4$$ In this way, we end up with four solutions, corresponding to different choices for the two free variables.