We have:

(1) x²yz = 12      (2) xy²z = 6       (3) xyz² = 18

Solve each equation for z.

z = 12/(x²y) = 6/(xy²) = sqrt(18)/sqrt(xy)

One proportion leads to: 12xy² = 6x²y

This simplifies to: y = x/2

Another proportion leads to: 6 sqrt(xy) = sqrt(18) xy²

This simplifies to: y³ = 2/x

By substitution we have: (x/2)³ = 2/x

This simplifies to: x⁴ = 16 <=> x = 2
Case x = 2:  Back substitute to find y = 1 and z = 3
Case x = -2: Back substitute to find y = -1 and z = -3

Therefore: x + y + z = 6

View different answers : radagast | Soarer|Soroban