To solve this Linear Programming Problem (LPP), we will:

1. Identify the constraints.
2. Plot the feasible region.
3. Determine the corner points of the feasible region.
4. Evaluate the objective function $z = 3x + 9y$ at each corner point.
5. Identify the maximum value of $z$.

Constraints:

1. $2y - x \leq 2$
2. $3y - x \geq 0$
3. $2x + 3y \leq 10$
4. $x \geq 0$, $y \geq 0$

Let's calculate step by step.import numpy as np import matplotlib.pyplot as plt from scipy.optimize import linprog

Coefficients of the objective function z = 3x + 9y

c = [-3, -9] # Negated for maximization

Coefficients of the constraints (Ax <= b)

A = [ [-1, 2], # 2y - x <= 2 [-1, 3], # 3y - x >= 0 -> -3y + x <= 0 [2, 3] # 2x + 3y <= 10 ]

b = [2, 0, 10]

Bounds for x and y (x >= 0, y >= 0)

x_bounds = (0, None) y_bounds = (0, None)

Solve the linear programming problem

result = linprog(c, A_ub=A, b_ub=b, bounds=[x_bounds, y_bounds], method='highs')

Extract the results

optimal_value = -result.fun # Negated back to get the maximum value optimal_point = result.x # Optimal values for x and y

optimal_value, optimal_point