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question about Solve

Dear All,

I don't understand the following behaviour of Solve.

Consider the following system :


Everything fine, I obtain :

{{z -> 0.1432394487827058/(f*i),
    x -> (1.419827298426263*^-9*(9.83403688*^9*a +
              1.966807376*^10*b)*e*i)/((c + 2.*d)*g*h),
    y -> (0.0716197243913529*g*h)/(e*i)}}

But if I ask for the answer of almost the same (only a 4 in the denominator
of the second equation), Solve isn't abble anymore to manage without using
inverse functions... why?

Solve[{x*y == (a + 2*b)/(c + 2*d),
 1/Sqrt[2] == Sqrt[(e*y)/(4*z*f*g*h)],
    2*Pi*i == 0.9/(z*f)}, {x, y, z}]


If I have numerical values for a, b, c, d, e, f, g, h, i:
a = 65/10^6;
b = 1/10^3;
c = 1.9;
d = 0.19;
e = 1/(2.5/10^3);
v = 18;
w = 8;
g = (2*v)/((c + 2*d)*w);
i = 3000;
h = 0.2;

Then Solve isn't able anymore! Mathematica thinks there is no solution. But
there is one. I have to use Reduce or give the numerical values as rules
after the Solve to find them.

In[247]:= Solve[{x*y == (a + 2*b)/(c + 2*d),
 1/Sqrt[2] == Sqrt[(e*y)/(z*f*g*h)],
    2*Pi*i == 0.9/(z*f)}, {x, y, z}]

Out[247]= {}

Can somebody explain this?

Thanks in advance,



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