1) Solve max {7x + 3y }; subject to constraints:

2x + 5y ≤ 28; 8x + 3y ≤ 48; x,y ≥ 0, integer.

Plot the feasible region first to help identify all integer solutions.

2) Let x₁ = 2, x₂ = 0, x₃ = 4 is an optimal solution to the linear program

max {4x₁ + 2x₂ + 3x₃ }; subject to constraints:

2x₁ + 3x₂ + x₃ ≤ 12; x₁ + 4x₂ + 2x₃ ≤ 10; 3x₁ + x₂ + x₃ ≤ 10; x₁, x₂, x₃ ≥ 0.

Applying the duality theory, find an optimal solution to the dual problem.

**Subject Mathematics Operations Research**