If $$\frac{x}{{\left( {2x + y + z} \right)}}$$   = $$\frac{y}{{\left( {x + 2y + z} \right)}}$$   = $$\frac{z}{{\left( {x + y + 2z} \right)}}   = a{\text{,}}$$     then find a, If x + y + z ≠ 0

$$\eqalign{ & \frac{x}{{\left( {2x + y + z} \right)}} = a \cr & \Rightarrow x = a\left( {2x + y + z} \right)\,....(1) \cr & \frac{y}{{\left( {x + 2y + z} \right)}} = a \cr & \Rightarrow y = a\left( {x + 2y + z} \right)\,....(2) \cr & \frac{z}{{\left( {x + y + 2z} \right)}} = a \cr & \Rightarrow z = a\left( {x + y + 2z} \right)\,....(3) \cr & {\text{Adding (1), (2) and (3) we get:}} \cr & x + y + z \cr & = a\left( {4x + 4y + 4z} \right) \cr & \Rightarrow a = \frac{{\left( {x + y + z} \right)}}{{4\left( {x + y + z} \right)}} \cr & \Rightarrow a = \frac{1}{4} \cr} $$