https://file.aiquickdraw.com/mathbot/file/1730895788904et8sx9hw.jpg

The image shows a set of linear programming (LP) problems where each one has an objective function to maximize (Max Z) and a set of constraints. Here is a structured breakdown of each problem:

1. **Problem 1:**
   - Objective: Maximize \( Z = 3X_1 + 3X_2 \)
   - Constraints:
     - \( 2X_1 + X_2 \leq 10 \)
     - \( 3X_1 + 4X_2 \geq 72 \)
     - \( X_1 \geq 0 \), \( X_2 \geq 0 \)

2. **Problem 2:**
   - Objective: Maximize \( Z = 3X_1 + X_2 \)
   - Constraints:
     - \( 2X_1 - X_2 \leq 10 \)
     - \( X_1 \leq 20 \)
     - \( X_1 \geq 0 \), \( X_2 \geq 0 \)

3. **Problem 3:**
   - Objective: Maximize \( Z = 5X_1 - 2X_2 \)
   - Constraints:
     - \( 2X_1 - X_2 \leq 10 \)
     - \( X_1 \leq 20 \)
     - \( X_1 \geq 0 \), \( X_2 \geq 0 \)

4. **Problem 4:**
   - Objective: Maximize \( Z = 6X_1 + 12X_2 \)
   - Constraints:
     - \( X_1 + 2X_2 \leq 50 \)
     - \( 2X_1 + X_2 \leq 40 \)
     - \( X_1 \geq 0 \), \( X_2 \geq 0 \)

5. **Problem 5:**
   - Objective: Maximize \( Z = 2X_1 + 3X_2 \)
   - Constraints:
     - \( 4X_1 + 5X_2 \leq 100 \)
     - \( 2X_1 + 4X_2 \leq 80 \)
     - \( X_1 \geq 0 \), \( X_2 \geq 0 \)

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To solve these problems, you can use both the **graphical method** (suitable for two-variable problems) and **Simplex method** (which is commonly used for more complex or larger LP problems).

Would you like a detailed solution for any specific problem or an explanation of the Simplex method? 

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### Related Questions:
1. What is the graphical method, and how does it apply to these problems?
2. How can the Simplex method be used to solve these linear programming problems?
3. What are the differences in results between graphical and Simplex methods?
4. Can these problems be solved using software tools, and if so, which ones are recommended?
5. How can constraints impact the feasibility of a solution in linear programming?

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### Tip:
When dealing with linear programming, always check if a feasible solution exists by analyzing constraints, as some combinations may lead to infeasibility or unbounded solutions.

Solve the following linear programming problems using graphical and Simplex methods.

Learn how to solve linear programming problems with two variables using graphical and Simplex methods. The problems include maximizing objective functions with constraints, exploring feasibility and optimality conditions. Suitable for undergraduate students studying operations research and optimization.

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