I have the following relatively large system of nonlinear equations over $GF(2)$:
$ 0 = w_7x_7 + w_7x_5 + w_7x_4 + w_7x_0 + w_6x_6 + w_6x_5 + w_6x_1 + w_5x_7 + w_5x_6 + w_5x_2 + w_4x_7 + w_4x_3 + w_3x_4 + w_2x_5 + w_1x_6 + w_0x_7 \\ 0 = w_7x_6 + w_7x_4 + w_7x_3 + w_6x_7 + w_6x_5 + w_6x_4 + w_6x_0 + w_5x_6 + w_5x_5 + w_5x_1 + w_4x_7 + w_4x_6 + w_4x_2 + w_3x_7 + w_3x_3 + w_2x_4 + w_1x_5 + w_0x_6 \\ 0 = w_7x_5 + w_7x_3 + w_7x_2 + w_6x_6 + w_6x_4 + w_6x_3 + w_5x_7 + w_5x_5 + w_5x_4 + w_5x_0 + w_4x_6 + w_4x_5 + w_4x_1 + w_3x_7 + w_3x_7 + w_3x_6 + w_3x_2 + w_2x_7 + w_2x_3 + w_1x_4 + w_0x_5 \\ 0 = w_7x_7 + w_7x_4 + w_7x_2 + w_7x_1 + w_6x_5 + w_6x_3 + w_6x_2 + w_5x_6 + w_5x_4 + w_5x_3 + w_5x_3 + w_4x_7 + w_4x_5 + w_4x_4 + w_4x_0 + w_3x_6 + w_3x_5 + w_3x_1 + w_2x_7 + w_2x_6 + w_2x_2 + w_1x_7 + w_1x_3 + w_0x_4 \\ 0 = w_7x_7 + w_7x_6 + w_7x_5 + w_7x_4 + w_7x_3 + w_7x_1 + w_6x_7 + w_6x_6 + w_6x_5 + w_6x_4 + w_6x_2 + w_5x_7 + w_5x_6 + w_5x_5 + w_5x_3 + w_4x_7 + w_4x_6 + w_4x_4 + w_3x_7 + w_3x_5 + w_3x_0 + w_2x_6 + w_1x_7 + w_1x_2 + w_0x_3 \\ 0 = w_7x_6 + w_7x_3 + w_7x_2 + w_6x_7 + w_6x_4 + w_6x_3 + w_5x_5 + w_5x_4 + w_4x_6 + w_4x_5 + w_3x_7 + w_3x_6 + w_2x_7 + w_2x_0 + w_1x_1 + w_0x_2 \\ 0 = w_7x_7 + w_7x_5 + w_7x_2 + w_7x_1 + w_6x_6 + w_6x_3 + w_6x_2 + w_5x_7 + w_5x_4 + w_5x_3 + w_4x_5 + w_4x_4 + w_3x_6 + w_3x_5 + w_2x_7 + w_2x_6 + w_1x_7 + w_1x_0 + w_0x_1 \\ 1 = w_7x_6 + w_7x_5 + w_7x_1 + w_6x_7 + w_6x_6 + w_6x_2 + w_5x_7 + w_5x_3 + w_4x_4 + w_3x_5 + w_2x_6 + w_1x_7 + w_0x_0 \\ $
I would like to solve the system for the 8 $w$ variables in terms of $x$. I know Sage and Magma implement relatively new techniques for converting this system to its CNF equivalent and then solving using popular SAT solvers like Minisat, but such techniques do not allow one to "fix" (for lack of a better word) variables (i.e. they solve the solve for exact values of $w0\dots w7$ and $x0 \dots x7$).
Does anyone know how I can accomplish this task using a CAS? The system above is really only a small example; I have larger systems with more variables whose equations have more monomials, so doing this task by hand becomes very time consuming.
In general you could use Groebner basis methods, but it turns out that your system is much easier: it's linear in $w$.
(although you may still want to solve it using Groebner basis methods even though it's linear so as to easily keep things reduced modulo the relations like $x_7^2 + x_7$)