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**Quiz #3: Linear Programming: Simplex Method**

Maximize:  
\[ P = 10x + 25y \]

Subject to:  
\[ 3x + 2y \leq 21 \]  
\[ x + 5y \leq 15 \]  
\[ x, y \geq 0 \]

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This is a typical linear programming problem where the objective is to maximize \( P \) (the profit function), subject to the given constraints.

Would you like me to solve this problem using the **Simplex Method** and provide step-by-step details? Or would you like help on any specific part of this?

Here are five follow-up questions that might help you deepen your understanding:

1. How is the initial simplex tableau set up for this problem?
2. What is the role of slack variables in the Simplex Method?
3. How do we decide which variable enters and leaves the basis in each iteration?
4. Can the Simplex Method handle problems with more complex constraints, such as equality or unbounded solutions?
5. What happens if the problem has alternate optimal solutions?

**Tip:** When solving linear programming problems, always double-check if the solution satisfies all the constraints before concluding the final answer.

Maximize: P = 10x + 25y subject to the constraints 3x + 2y ≤ 21, x + 5y ≤ 15, and x, y ≥ 0.

Learn how to solve a linear programming problem using the Simplex Method. Maximize P = 10x + 25y under constraints 3x + 2y ≤ 21, x + 5y ≤ 15, and x, y ≥ 0. A step-by-step guide for optimization problems suitable for advanced high school and college-level students.

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