Maximize Z = 10x1 + 20x2
Subject to
-x1 + 2x2 ≤ 15
x1 + x2 ≤ 12
5x1 + 3x2 ≤ 45
And
X1 ≥ 0, x2 ≥ 0.
Solve the problem above by:
Using the graphical method
-x1 + 2x2 = 15
X1 +x2 = 12
5x1 + 3x2 = 45

could you show me in a graph witch area is correct

To solve the given linear programming problem using the graphical method, we will follow these steps:

1. **Plot the constraints on a graph:**
   - Each inequality represents a half-plane, and we’ll convert them into equations to find the boundary lines.
   - The lines are:
     - $ -x_1 + 2x_2 = 15 $
     - $ x_1 + x_2 = 12 $
     - $ 5x_1 + 3x_2 = 45 $

2. **Identify the feasible region:**
   - The feasible region is the area where all the inequalities are satisfied simultaneously. This is the intersection of the half-planes defined by each constraint.

3. **Determine the optimal solution:**
   - The objective function $ Z = 10x_1 + 20x_2 $ will be maximized at a corner point (vertex) of the feasible region.

Let's now proceed to graph these lines and determine the feasible region, which will include the vertices. I'll then provide the graphical representation for you. numpy as np
import matplotlib.pyplot as plt

# Define the constraint lines as functions of x1
x1_values = np.linspace(0, 15, 400)

# Constraint 1: -x1 + 2x2 = 15 => x2 = (x1 - 15) / 2
x2_1 = (x1_values - 15) / 2

# Constraint 2: x1 + x2 = 12 => x2 = 12 - x1
x2_2 = 12 - x1_values

# Constraint 3: 5x1 + 3x2 = 45 => x2 = (45 - 5x1) / 3
x2_3 = (45 - 5 * x1_values) / 3

# Plot the constraint lines
plt.figure(figsize=(8, 6))

plt.plot(x1_values, x2_1, label=r'$-x_1 + 2x_2 = 15$', color='blue')
plt.plot(x1_values, x2_2, label=r'$x_1 + x_2 = 12$', color='green')
plt.plot(x1_values, x2_3, label=r'$5x_1 + 3x_2 = 45$', color='red')

# Add shading to the feasible region
plt.fill_between(x1_values, np.maximum(0, x2_1), np.minimum(x2_2, x2_3), where=(x1_values <= 12) & (x1_values >= 0), 
                 color='gray', alpha=0.5)

# Adding constraints for non-negativity of x1 and x2
plt.xlim(0, 12)
plt.ylim(0, 12)

# Adding labels and title
plt.xlabel(r'$x_1$')
plt.ylabel(r'$x_2$')
plt.title('Feasible Region for the LP Problem')

# Add a legend
plt.legend()

# Show the graph
plt.grid(True)
plt.show()The graph above represents the constraints and the feasible region of the linear programming problem.

- **The blue line** represents the constraint $ -x_1 + 2x_2 = 15 $.
- **The green line** represents the constraint $ x_1 + x_2 = 12 $.
- **The red line** represents the constraint $ 5x_1 + 3x_2 = 45 $.

The shaded area represents the feasible region where all the inequalities are satisfied. The region of interest lies within this shaded area, and it will be bounded by the intersection points of these lines.

Next steps:
- The **corner points** of this feasible region (vertices of the shaded area) will be where we will evaluate the objective function $ Z = 10x_1 + 20x_2 $ to find the maximum value.
- We should now find the exact coordinates of the corner points to evaluate the objective function.

Would you like me to calculate these corner points and evaluate the objective function at each one?

Maximize Z = 10x1 + 20x2 subject to -x1 + 2x2 ≤ 15, x1 + x2 ≤ 12, 5x1 + 3x2 ≤ 45, and x1 ≥ 0, x2 ≥ 0. Solve the problem above by using the graphical method. Could you show me in a graph which area is correct?

Learn how to solve a linear programming problem using the graphical method. This solution maximizes Z = 10x1 + 20x2 subject to given constraints, using graphical techniques to identify the feasible region and optimal solution. Suitable for high school students studying linear programming.

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