 Solution

Both objectives equal $1/4$: $1^{\mathsf{T}}x=1/12+1/6=1/4$ and $1^{\mathsf{T}}y=1/6+1/12=1/4$. Weak duality bounds the value of every primal feasible point above by every dual feasible objective. Equality for this feasible pair therefore proves both are optimal, without needing to solve either problem further.