how to solve a system of equations with inexact coefficients

I going to ask my question in another way, and excuse me for my English. My question is

  H1[x_, y_]=(d x + w y) + 2(c x - a y)/d + y^2 a^2 

I need to chose values for those five parameters for some reason, then, for example, I choose them like

S1 = {b -> -1.2, c -> 0.1, d -> 4, a -> - 0.8, w -> 2} 

When I try to search for the intersection values of H1 with the y axis

Solve[(H1[0, y] /. S1) == -0.5, y]  (*{{y -> -3.52859}, {y -> -0.221405}}*)  y1 = -3.5285945694153686; y2 = -0.22140543058463089; Solve[(H1[0, y] /. S1) == -1, y] (*{{y -> -3.27254}, {y -> -0.477458}}*)   y3 = -3.272542485937368;  y4 = -0.4774575140626314; 

now I have

H2[x_, y_] = (-1 + 4 (c1 + b1 y)^2 - 2 (1 + x α1 + γ1)^3 + a1 x + y β1)/(c1 + b1 y)^3 

another function with six new parameters, i need to find these parameters by solving the system

eq1 = Numerator[Factor[H2[0, y1] - H2[0, y2]]] eq2 = Numerator[Factor[H2[0, y3] - H2[0, y4]]] 

This system is with inexact parameters

Sol = Solve[({eq1, eq2} /. {b1 -> 1, γ1 -> 0.5}) == 0, {β1, c1}, Reals] 

When i use Solve, it’s giving me a solution of the same system but with inexact coefficients. My problem is when I try to solve this system

e1=H1[0, y]-H1[0, Y] e2=H2[0, y]-H2[0, Y] 

I need to find the same results of y1, y2, y3 and y4, but since Solve uses exact parameters, I don’t find the same results. I need a solution, please, thank you.