Solve the LPP graphically. Minimize Z = 3x + 5y, Subject to constraints

• 2x + 3y ≥ 12
• -x + y ≤ 3
• x ≤ 4
• y ≥ 3

To solve the linear programming problem (LPP) graphically, we need to minimize the objective function Z = 3x + 5y under the given constraints. Here’s how to approach it step by step:

Step 1: Identify the Constraints

The constraints provided are:

• 2x + 3y ≥ 12
• -x + y ≤ 3
• x ≤ 4
• y ≥ 3

Step 2: Convert Inequalities to Equations

To graph the constraints, convert each inequality into an equation:

• 2x + 3y = 12
• -x + y = 3
• x = 4
• y = 3

Step 3: Graph the Lines

Plot each line on a graph:

• For 2x + 3y = 12, find intercepts: (6, 0) and (0, 4).
• For -x + y = 3, find intercepts: (0, 3) and (3, 0).
• For x = 4, draw a vertical line at x = 4.
• For y = 3, draw a horizontal line at y = 3.

Step 4: Determine the Feasible Region

Shade the appropriate areas based on the inequalities:

• Above the line 2x + 3y = 12.
• Below the line -x + y = 3.
• Left of the line x = 4.
• Above the line y = 3.

The feasible region is where all shaded areas overlap.

Step 5: Identify Corner Points

Find the coordinates of the corner points of the feasible region. These points are where the lines intersect:

• Intersection of 2x + 3y = 12 and -x + y = 3.
• Intersection of 2x + 3y = 12 and x = 4.
• Intersection of -x + y = 3 and y = 3.
• Intersection of x = 4 and y = 3.

Step 6: Evaluate the Objective Function

Calculate Z = 3x + 5y at each corner point:

• At (0, 4): Z = 3(0) + 5(4) = 20
• At (4, 0): Z = 3(4) + 5(0) = 12
• At (4, 3): Z = 3(4) + 5(3) = 27
• At (3, 3): Z = 3(3) + 5(3) = 24

Step 7: Find the Minimum Value

Compare the values of Z calculated at each corner point. The minimum value occurs at:

Minimum Z = 12 at the point (4, 0).

This graphical method allows you to visualize the constraints and find the optimal solution effectively.