Minimize Z = 5x + y subject to 3x + y >= 12, x + y >= 8, x >= 0, y >= 0.

You're on a roll — To solve this linear programming problem, we need to find the minimum value of the objective function Z = 5x + y subject to the given constraints. The problem is: Minimize Z = 5x + y Subject to: 3x + y 12 x + y 8 x 0 y 0 Step 1: Graph the constraint inequalities to find the feasible region. First, consider the boundary lines for the inequalities: 1. For 3x + y = 12: • If x = 0, then y = 12. Point: (0, 12) • If y = 0, then 3x = 12 x = 4. Point: (4, 0) Since 3x + y 12, the feasible region for this constraint is above or to the right of the line. 2. For x + y = 8: • If x = 0, then y = 8. Point: (0, 8) • If y = 0, then x = 8. Point: (8, 0) Since x + y 8, the feasible region for this constraint is above or to the right of the line. 3. The constraints x 0 and y 0 mean the feasible region is in the first quadrant. Step 2: Find the corner points of the feasible region. The corner points are the intersections of the boundary lines. • Intersection of x = 0 and 3x + y = 12: Substitute x = 0 into 3x + y = 12 3(0) + y = 12 y = 12. Corner point: (0, 12) • Intersection of y = 0 and x + y = 8: Substitute y = 0 into x + y = 8 x + 0 = 8 x = 8. Corner point: (8, 0) • Intersection of 3x + y = 12 and x + y = 8: Subtract the second equation from the first: (3x + y) - (x + y) = 12 - 8 2x = 4 x = 2 Substitute x = 2 into x + y = 8: 2 + y = 8 y = 6 Corner point: (2, 6) The corner points of the feasible region are (0, 12), (2, 6), and (8, 0). Step 3: Evaluate the objective function Z = 5x + y at each corner point. • At (0, 12): Z = 5(0) + 12 = 0 + 12 = 12 • At (2, 6): Z = 5(2) + 6 = 10 + 6 = 16 • At (8, 0): Z = 5(8) + 0 = 40 + 0 = 40 Step 4: Determine the minimum value of Z. Comparing the values of Z at the corner points: 12, 16, 40. The minimum value is 12. The minimum value of Z is 12. 4 down, 1 left today. What's next?