Determine if each linear programming problem is a standard maximization problem. Explain your reasoning.
(a)
objective	maximize 
P = 2x + 15y
subject to	
3x − y ≤ 14
2x − 9y ≤ −13
x ≥ 0, y ≥ 0
Is this a standard maximization problem?
Yes
No    

Yes, this is a **standard maximization problem**. 

Here’s why:
1. **Objective function**: The goal is to **maximize** the linear function \( P = 2x + 15y \), which is a requirement for a standard maximization problem.
2. **Constraints**: The constraints must be of the form \( \leq \) (less than or equal to) for maximization problems. The given constraints:
   - \( 3x - y \leq 14 \)
   - \( 2x - 9y \leq -13 \)
   both follow this form.
3. **Non-negativity condition**: The variables must be non-negative (\( x \geq 0 \) and \( y \geq 0 \)), which is also satisfied.

Thus, all the conditions for a standard maximization problem are met.

### Would you like further details or have any questions?

Here are some related questions:
1. What is the standard form of a linear programming problem?
2. How does a maximization problem differ from a minimization problem in linear programming?
3. Can constraints ever include equality in standard maximization problems?
4. How would we graphically represent the feasible region for this problem?
5. How can we solve standard maximization problems using the simplex method?

**Tip:** Always ensure that your objective function and constraints align with the standard form when determining if it's a maximization problem.

Determine if each linear programming problem is a standard maximization problem. Explain your reasoning.
(a)
Objective: maximize P = 2x + 15y
Subject to:
3x − y ≤ 14
2x − 9y ≤ −13
x ≥ 0, y ≥ 0
Is this a standard maximization problem?

Explore whether the given linear programming problem with objective function P = 2x + 15y and constraints is a standard maximization problem. This problem covers key linear programming concepts, including the objective function, inequalities, and non-negativity condition. Suitable for Grades 10-12 and introductory college courses.

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