Solve the following linear programming problem graphically:

Maximize Z = 50x + 15y

Subject to

• 5x + y ≤ 100
• x + y ≤ 60
• x, y ≥ 0

To solve the linear programming problem graphically, we need to maximize the objective function Z = 50x + 15y while adhering to the given constraints. Here’s how to approach it step by step:

Step 1: Identify the Constraints

The constraints provided are:

• 5x + y ≤ 100
• x + y ≤ 60
• x ≥ 0
• y ≥ 0

Step 2: Graph the Constraints

To graph these inequalities, we first convert them into equations:

• 5x + y = 100
• x + y = 60

Next, we find the intercepts for each line:

• For 5x + y = 100:
  • x-intercept: (20, 0)
  • y-intercept: (0, 100)
• For x + y = 60:
  • x-intercept: (60, 0)
  • y-intercept: (0, 60)

Step 3: Plot the Lines

On a graph, plot the lines based on the intercepts. The area of interest is where the shaded regions of the inequalities overlap, which represents the feasible region.

Step 4: Identify the Corner Points

The feasible region is bounded by the lines and the axes. The corner points (vertices) of this region can be found by solving the equations:

• Intersection of 5x + y = 100 and x + y = 60
• Intersection of 5x + y = 100 and the x-axis (y = 0)
• Intersection of x + y = 60 and the x-axis (y = 0)
• Intersection of x + y = 60 and the y-axis (x = 0)

Step 5: Calculate the Intersection Points

Solving the equations:

• From 5x + y = 100 and x + y = 60, we find the intersection point (8, 52).
• From 5x + y = 100 and the x-axis, we find (20, 0).
• From x + y = 60 and the x-axis, we find (60, 0).
• From x + y = 60 and the y-axis, we find (0, 60).

Step 6: Evaluate the Objective Function

Now, we evaluate Z = 50x + 15y at each corner point:

• At (8, 52): Z = 50(8) + 15(52) = 400 + 780 = 1180
• At (20, 0): Z = 50(20) + 15(0) = 1000
• At (60, 0): Z = 50(60) + 15(0) = 3000
• At (0, 60): Z = 50(0) + 15(60) = 900

Step 7: Determine the Maximum Value

The maximum value of Z occurs at the point (60, 0) with Z = 3000. Therefore, the optimal solution to the linear programming problem is:

Max Z = 3000 at (x, y) = (60, 0)