Maximize Z = 5x + 3y, subject to 3x + 5y ≤ 15, 5x + 2y ≤ 10, 
x ≥ 0, y ≥ 0. 


To solve this linear programming problem, we need to determine the values of $x$ and $y$ that maximize $Z = 5x + 3y$ while satisfying the constraints:

1. $3x + 5y \leq 15$
2. $5x + 2y \leq 10$
3. $x \geq 0, y \geq 0$

### Steps to Solve:

#### 1. Identify the constraints and plot the feasible region
Each inequality represents a line in the $xy$-plane. The feasible region is the intersection of the half-planes defined by these inequalities, bounded by $x \geq 0$ and $y \geq 0$.

#### 2. Find the corner points
The optimal solution lies at one of the corner points of the feasible region. These points are where the boundary lines intersect or where they meet the axes.

#### 3. Evaluate $Z = 5x + 3y$ at each corner point
Substitute the coordinates of each corner point into $Z$ and find the maximum value.

Let me calculate and display the solution graphically and numerically.import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import linprog

# Coefficients for the objective function
c = [-5, -3]  # Coefficients for Z = 5x + 3y (minimizing -Z to maximize Z)

# Coefficients of inequality constraints
A = [[3, 5], [5, 2]]
b = [15, 10]

# Bounds for x and y
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 results
optimal_x, optimal_y = result.x
max_z = -result.fun  # Convert back to maximization

# Create plot to visualize the feasible region
x = np.linspace(0, 5, 400)
y1 = (15 - 3 * x) / 5  # Line for 3x + 5y = 15
y2 = (10 - 5 * x) / 2  # Line for 5x + 2y = 10

plt.figure(figsize=(8, 6))
plt.plot(x, y1, label=r"$3x + 5y \leq 15$")
plt.plot(x, y2, label=r"$5x + 2y \leq 10$")
plt.axhline(0, color='black',linewidth=0.5)
plt.axvline(0, color='black',linewidth=0.5)
plt.fill_between(x, np.minimum(y1, y2), 0, where=(y1 >= 0) & (y2 >= 0), color='gray', alpha=0.3)
plt.scatter(optimal_x, optimal_y, color='red', label='Optimal Solution', zorder=5)
plt.xlim(0, 5)
plt.ylim(0, 5)
plt.xlabel("x")
plt.ylabel("y")
plt.title("Feasible Region and Optimal Solution")
plt.legend()
plt.grid()

plt.show()

optimal_x, optimal_y, max_z
<<ImageDisplayed>>(1.0526315789473688, 2.3684210526315788, 12.36842105263158)(1.0526315789473688, 2.3684210526315788, 12.36842105263158)successThe optimal solution to maximize $ Z = 5x + 3y $ under the given constraints is:

- $ x = 1.05 $ (approximately)
- $ y = 2.37 $ (approximately)
- Maximum value of $ Z = 12.37 $ (approximately)

The red dot in the graph represents this optimal solution.

Would you like further details or additional analysis?

Here are 5 related questions for exploration:
1. What happens if one of the constraints changes, such as $ 5x + 2y \leq 12 $?
2. How do you solve this problem using the simplex method step by step?
3. What are slack variables, and how would they apply here?
4. Can this problem be solved graphically for all cases, and what are its limitations?
5. How would introducing a third variable $ z $ impact the problem?

**Tip:** Linear programming problems can often be visualized effectively when there are two variables, making it easier to understand the feasible region.

Maximize Z = 5x + 3y, subject to 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0.

Learn how to solve a linear programming problem to maximize Z = 5x + 3y under constraints 3x + 5y ≤ 15, 5x + 2y ≤ 10, and x, y ≥ 0. Explore graphical and numerical methods, with an optimal solution at x ≈ 1.05, y ≈ 2.37, and Z ≈ 12.37. Suitable for high school students in Grades 10-12.

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