I have to solve an overdetermined system of non-linear equations. My system has a lot of equations, but here, for example, my system has $4$ equations. Because all the variables have to be binary, I have to add equations like $x_i^2-x_i$. Now I have $8$ equations with $4$ variables. This system is
$$\begin{aligned} (1-x)(1-y)(1-z)(1-t) &= 0 \\ (1-x)(1-y) &=0 \\ (1-y)(1-z)(1-t) &=0 \\ (1-x)(1-z)(1-t) &=0 \end{aligned} \\ x^2 - x = 0,~ y^2-y=0,~z^2-z=0,~t^2-t=0.$$
I know in advance, this system will have solution but I don't know how to explain it mathematically. (I know how to solve this system)
How can it be explained, because it is over-determined? What is the criterion? Any reference?