syms x y z
eqn = [0.0425*x*z^-2-0.7*x-0.00253*x*z+0.00134*x*z^2-0.03*x*z^1.7+0.0063*x*z^3.4-0.1*x*z^0.07+0.00047*x*z^2.7+0.18*y*z^-2-0.865*y-0.013*y*z+0.0002*y*z^2-0.03*y*z^1.7+0.0019*y*z^3.4-0.5*y*z^0.7+0.00023*y*z^2.7==0
and wait, and wait, and wait.... Hope you have a lot of memory.
Your question has obvious 4th powers of z, and has decimal exponents such as ^1.7 and ^4.4 . If you are looking for all solutions in exact closed-form representation then you are going to have over 400 complex roots when the solution is eventually found, after the decimal exponents are rewritten in rational form.
If you just want one solution, in decimal form, then above use vpasolve() instead of solve()