To solve the given linear programming problem graphically:

Objective Function: Maximize ( Z = 10x + 15y )

Constraints:

1. ( 3x + 2y \leq 50 )

2. ( x + 4y \leq 20 )

3. ( x \leq 8 )

4. ( x, y \geq 0 )

Steps to Solve:

1. Plot the Constraints on a Graph

• Rewrite each inequality as an equation to find boundary lines: ( 3x + 2y = 50 ): When ( x = 0, y = 25 ). When ( y = 0, x = \frac{50}{3} \approx 16.67 ). ( x + 4y = 20 ): When ( x = 0, y = 5 ). When ( y = 0, x = 20 ). ( x = 8 ): A vertical line at ( x = 8 ).

• Plot these lines on the graph and shade the feasible region satisfying all constraints.

2. Identify Corner Points of the Feasible Region The feasible region is bounded by the intersection points of the constraints:

• Intersection of ( 3x + 2y = 50 ) and ( x + 4y = 20 ): Solve simultaneously: From ( x + 4y = 20 ), ( y = (20 - x)/4 ). Substitute into ( 3x + 2((20 - x)/4) = 50 ): Solve to get ( x = 8, y = 3 ).

• Other corner points are: ( (0,0) ), ( (8,0) ), and intersection of ( x + 4y = 20 ) with ( y=0: (20,0)).