Maximize Z = 3x + 5y
Subject to:
x, y >= 0
x + y <= 4

To solve this linear programming problem, we can use the graphical method since we have two decision variables. Here's the detailed process:

### Objective Function
Maximize \( Z = 3x + 5y \)

### Constraints
1. \( x \geq 0 \)
2. \( y \geq 0 \)
3. \( x + y \leq 4 \)

### Steps to Solve

1. **Graph the Constraints**: Plot the feasible region defined by the constraints on the \(xy\)-plane.

2. **Identify the Feasible Region**: This is the area where all the constraints overlap.

3. **Find the Corner Points**: The maximum or minimum values for the objective function will occur at the vertices (corner points) of the feasible region.

4. **Evaluate the Objective Function at Each Corner Point**: Calculate the value of the objective function \( Z \) at each of these points.

### Graphing the Constraints

1. **Plot \( x \geq 0 \) and \( y \geq 0 \)**: These constraints indicate that the feasible region is in the first quadrant.

2. **Plot \( x + y \leq 4 \)**:
   - When \( x = 0 \), \( y = 4 \)
   - When \( y = 0 \), \( x = 4 \)
   - Draw a line passing through the points \( (0, 4) \) and \( (4, 0) \)

### Identify the Feasible Region
The feasible region is a triangle formed by the points \( (0, 0) \), \( (4, 0) \), and \( (0, 4) \).

### Find the Corner Points
The corner points (vertices) of the feasible region are:
1. \( (0, 0) \)
2. \( (4, 0) \)
3. \( (0, 4) \)

### Evaluate the Objective Function at Each Corner Point

1. \( Z(0, 0) = 3(0) + 5(0) = 0 \)
2. \( Z(4, 0) = 3(4) + 5(0) = 12 \)
3. \( Z(0, 4) = 3(0) + 5(4) = 20 \)

### Conclusion
The maximum value of \( Z \) occurs at the point \( (0, 4) \), and the maximum value is \( Z = 20 \).

### Summary
- **Objective Function**: \( Z = 3x + 5y \)
- **Constraints**:
  - \( x \geq 0 \)
  - \( y \geq 0 \)
  - \( x + y \leq 4 \)
- **Maximum Value**: \( Z = 20 \) at \( (0, 4) \)

Would you like a graphical representation of this solution?

### Follow-up Questions
1. Do you have any additional constraints to add to this problem?
2. Would you like to see a step-by-step graphical solution?
3. Are there any specific methods (e.g., Simplex) you prefer for solving linear programming problems?
4. Do you need assistance with formulating more complex linear programming problems?
5. Are there any other objective functions or constraints you want to explore?
6. Would you like to learn more about the Simplex method for solving linear programming problems?
7. Do you need a detailed explanation of the feasibility region?
8. Are you interested in solving similar problems with different constraints or objective functions?

### Tip
When solving linear programming problems graphically, always ensure that all constraints are correctly plotted and that the feasible region is accurately identified. The optimal solution will always be at one of the vertices of the feasible region.

Explore how to maximize the objective function Z = 3x + 5y using the graphical method in linear programming. This problem involves plotting constraints, identifying the feasible region, and finding optimal solutions. Suitable for advanced high school students studying linear programming.

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