Help Fast!

Ok, first off, let's split the one equation into a system of equations so that each term is equal to another. We get

$x + 1/y = y + 1/z\\ x + 1/y = z + 1/x\\ y + 1/z = z + 1/x$

I won't really show the steps, although I will if needed. Let me know if you want to. 

$(x, y, z) = (-\frac{1}{z-1}, \frac{z-1}{z}, z)\\ (x, y,z) = (-\frac{1}{z+1}, -\frac{z+1}{z}, z)$

We want to find the value of xyz since we need the product. 

Thus, multiplying these together, we get

$(-\frac{1}{z-1})(\frac{z-1}{z})z=-\frac{1}{z-1}\left(z-1\right)=-1$

$(-\frac{1}{z+1} )(-\frac{z+1}{z}) z=\frac{1}{z}z=1$

So good job! You were correct about 1. The other value is -1. 

I didn't go into too much detail, but let me know if you need more assistance. 

Thanks! :)