Linear Programming Problem

(0, 0), (3, 0), (2, 2), (0, 2), (3, 1)

Corner points are intersections of boundary lines: Origin (0,0), x-axis intersections (3,0), y-axis intersections (0,2), and intersections of constraint lines like x+y=4 with x=3 giving (3,1), and x+y=4 with y=2 giving (2,2).

Decision variables, Objective function, Linear constraints, Non-negativity restrictions

An LPP requires: decision variables (x, y, etc.), an objective function to optimize (Z = ax + by), linear constraints (inequalities/equations), and non-negativity restrictions (x ≥ 0, y ≥ 0). Quadratic terms make it non-linear.

7

Corner points of feasible region: (1,3), (3,1), and intersection points. At (1,3): Z = 2(1) + 3(3) = 11. At (3,1): Z = 2(3) + 3(1) = 9. The point where x + y = 4 intersects the boundary gives minimum at (1,3) where Z = 7.

Variables take only non-negative values

Non-negativity constraints x ≥ 0, y ≥ 0 restrict the feasible region to the first quadrant only, ensuring variables cannot take negative values. This is essential for most real-world problems where negative quantities don't make sense.