Question

Solve the following linear programming problem graphically: 
Maximize \( z = x + y \), subject to constraints: 
\[ 2x + 5y \leq 100, \quad 8x + 5y \leq 200, \quad x \geq 0, \quad y \geq 0. \]

Hint:

To solve linear programming problems graphically, identify the feasible region by plotting all constraints and evaluate the objective function at each corner point.

Answer 1

Step 1: Plot the constraints

The given constraints are:

• \( 2x + 5y = 100 \): This line intersects the axes at \( (50, 0) \) and \( (0, 20) \).
• \( 8x + 5y = 200 \): This line intersects the axes at \( (25, 0) \) and \( (0, 40) \).
• \( x \geq 0,\; y \geq 0 \): These inequalities restrict the feasible region to the first quadrant.

The feasible region is the intersection of all the above constraints, forming a polygon.

Step 2: Determine corner points

The corner points of the feasible region are obtained by solving the constraints pairwise:

• \( O(0, 0) \): Intersection of \( x = 0 \) and \( y = 0 \).
• \( A(25, 0) \): Intersection of \( 8x + 5y = 200 \) and \( y = 0 \).
• \( B\left(\frac{50}{3}, \frac{40}{3}\right) \): Intersection of \( 2x + 5y = 100 \) and \( 8x + 5y = 200 \).
• \( C(0, 20) \): Intersection of \( 2x + 5y = 100 \) and \( x = 0 \).

Step 3: Evaluate the objective function

Substitute each corner point into the objective function \( z = x + y \):

Corner Point	Value of \( z = x + y \)
\( O(0, 0) \)	0
\( A(25, 0) \)	25
\( B\left(\frac{50}{3}, \frac{40}{3}\right) \)	30
\( C(0, 20) \)	20

Step 4: Identify the maximum value

The maximum value of \( z \) is 30, which occurs at \( B\left(\frac{50}{3}, \frac{40}{3}\right) \).

Conclusion:

\[ \boxed{z_{\max} = 30 \text{ at } \left(\frac{50}{3}, \frac{40}{3}\right)} \]