# 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.

• I reformatted your post, please, check it. Commented Aug 6, 2021 at 10:32
• Thank you very much. Commented Aug 6, 2021 at 10:41

Before substituting values you should derive the equation symbollically:

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


Evaluate the intersection x==0,y==ya

sol1[ya_] := y /. Solve[ya == H1[0, y], y]

H2[x_, y_] := (-1 + 4 (c1 + b1 y)^2 -2 (1 + x \[Alpha]1 + \[Gamma]1)^3 + a1 x +y \[Beta]1)/(c1 + b1 y)^3


Set of equations

eqn = {Numerator[Factor[H2[0, sol1[-1/2][[1]]] - H2[0, sol1[-1/2][[2]]]]],
Numerator[Factor[H2[0, sol1[-1 ][[1]]] - H2[0,sol1[-1 ][[2]]]]]};


Solve for special parameters:

subst = {b -> -1.2, c -> 0.1, d -> 4, a -> -0.8,w -> 2,b1 -> 1, \[Gamma]1 -> 0.5}
NSolve[eqn /. subst, {c1, \[Beta]1}, Reals]
(*{{c1 -> 1.07508, \[Beta]1 -> -11.2073},
{c1 -> -1.54514, \[Beta]1 ->9.84792},
{c1 -> 3.71031, \[Beta]1 -> -2.16057}}*)


Hope it helps!