Question

Help please:
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Solve the system of equation for positive real numbers:

(1/xy) = (x/z) + 1,(1/yz)=(y/x)+1,(1/zx)=z/y+1

Answer 1

assume x=y=z, with x,y, and z are real positive numbers
for the 1st equation, (1/xy) = (x/z) + 1 can rewrite be :
1/(x*x) = (x/x) + 1 or
1/x^2 = 1+1
1/x^2 = 2
x^2 = 1/2
x=+- (sqrt(1/2)) (but - sign not satisfied)
so, x=sqrt(1/2)
because x=y=z, obviously y=sqrt(1/2) , and z=sqrt(1/2) also

Answer 2

Thanks RadEn,it is working.However , how we can justify the assumption of x=y=z?