Let x, y, and z be distinct nonzero real numbers and

Question

x + (1/y) = y + (1/z) = z + (1/x)

Find all possible values of xyz.

I don't even know where to start!
Any help is appreciated, thank you :)

Answer 1

Just by inspection, it should be clear that
(x,y,z) = ±(1,1,1), ±(-1, 1/2, 2) are solutions
Then, set up the three equations
x + 1/y = y + 1/z
y + 1/z = z + 1/x
x + 1/y = z + 1/x
Some manipulation yields
y = -1/(x+1)
z = -(x+1)/x
so, pick any other values for x that you want.

Answer 2

Thanks for the response!

I still have a question though - what do you mean by picking any other values of x? Is there an equation that I'm supposed to solve?

Answer 3

I ran a computer simulation and the only solutions were of the type
(k,k,k), where k is any real number, k ≠ 0

but the values of (x,y,z) were to be distinct, so other than the above, there is no solution
e.g. oobleck has
y = -1/(x+1)
z = -(x+1)/x
but (2,2,2), which satisfies the original set of equations, does not satisfy this simplification.
I suspect an error in the manipulation

Answer 4

Thank you for the response.

Yes, I do think there might be a miscalculation. From running this through a calculator, I got y=1/(1-x).

Another thing: I don't get how one could get ±(-1, 1/2, 2) as a solution through just inspection. Is there math behind this?

Also, it appears that ±(-1, 1/2, 2) the only solution to the equation.

Answer 5

Oh, I've figured it out! Thanks for the help, but I've found another method.