The question doesn't assume that the equations are independent, which might add redundancy when solving them, and it doesn't assume that the equations are linear, in which case you are not guaranteed a solution at all.

Anyway, the equations are probably assumed implicitly to be linear, and you need to think in terms of free variables. If you are solving your system of linear equations, and you end up with, say, 7 free variables, then for each combination of choices of values for these free variables, the set of linear equations has a solution. But since each variable can take values in F_2 (giving 2 choices for each variable), this makes a total of 2^7 choices.