If x + 1/y = 3, y + 1/z = 2 and z + 1/x = 4, then find the value of xyz + 1/xyz?

Calculation:

x + 1/y = 3      ----(1) 

y + 1/z = 2      ----(2) 

z + 1/x = 4      ----(3)

Add the equation. (1), (2) and (3).

⇒ x + y + z + 1/x + 1/y + 1/z = 9      ----(4)

Now multiply the eq. (1), (2) and (3)

⇒ (x + 1/y) × (y + 1/z) × (z + 1/x) = 3 × 2 × 4

⇒ (xy + x/z + 1 + 1/zy)(z + 1/x) = 24

⇒ (xyz + y + x + 1/z + z + 1/x + 1/y + 1/xyz) = 24

⇒ [xyz + (1/xyz) + x + y + z + 1/x + 1/y + 1/z] = 24

⇒ xyz + 1/xyz + 9 = 24 

⇒ xyz + 1/xyz = 24 – 9 = 15

∴ The answer is 15