Non-linear system of eqtns

[Offroad]

Dear friends,

WOuld you be kind enough as to teach me how to solve systems of equations like this one and larger using the Prime?

z*y - x^2 + y = -53
x^2 + y^2 - z = 47
x*y - x + z = -11

initial guess would be {x,y,z}={1,5,0}

Thank you