Linearization of nonlinear equations in finite field

Small remark: all following should be considered in . Ie 1+1=0, x+x=0, 1+0=1, x+0=x,0+0=0 and so on. Field contains only {0,1}

Once again, working from example is the best. Ok, suppose we have set of equations:

It would be easily solved, like linear equations if it would of form x+y=something, but it doesnt work very well in case of xz+yw=something.

What to do? Well, lets change products to new variables then:

so we get from original set of equations:

This one is easily solved, resulting in a=1,b=1,c=1,d=0,e=1, therefore giving new set of equations:

It is also easily solved, quite trivial to see that and