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This problem is a linear programming problem, where the objective is to minimize the function \( z = 3x + 5y \) subject to the given constraints:

1. \( 2y + 5x \geq 18 \)
2. \( 6y + 3x \geq 27 \)
3. \( y + x \geq 6 \)
4. \( x \geq 0 \)
5. \( y \geq 0 \)

To solve this problem, we need to:

1. **Graph the constraints** to find the feasible region, which is the area that satisfies all the inequalities.
2. **Identify the corner points** of the feasible region. These are potential candidates for where the minimum value of \( z \) might occur.
3. **Calculate the value of \( z \) at each corner point** of the feasible region.
4. **Select the minimum value** of \( z \) from the calculated values.

Let's start by solving this step-by-step.

### Step 1: Graph the Constraints
Each inequality represents a half-plane. By graphing these inequalities, we find the region that satisfies all of them simultaneously.

### Step 2: Identify Corner Points
Find the points of intersection of the lines formed by the inequalities. These points, along with the boundary conditions \( x \geq 0 \) and \( y \geq 0 \), form the vertices of the feasible region.

### Step 3: Calculate \( z \) at Each Corner Point
Substitute each vertex into \( z = 3x + 5y \) to find its value.

### Step 4: Choose the Minimum \( z \)
Among the calculated values, the smallest one will be the minimum value of \( z \).

I'll go ahead and compute this.The linear programming problem has been solved successfully, and the results are as follows:

- **Minimum value of \( z \)**: 24.0
- **Optimal values of \( x \) and \( y \)**:
  - \( x = 3 \)
  - \( y = 3 \)

This means that the minimum value of \( z = 3x + 5y \) is 24, and it occurs at \( (x, y) = (3, 3) \).

Would you like more details or have any questions?

Here are some additional related questions you might find useful:
1. How do you graph the feasible region for this problem?
2. What are corner points, and why are they important in linear programming?
3. How can you use graphical methods to solve a linear programming problem?
4. What are some common applications of linear programming in real life?
5. What does the term "feasible region" mean in the context of linear programming?

**Tip:** For problems involving inequalities, always ensure the constraints are converted correctly before solving, especially when translating them into a form suitable for solvers like `linprog`.

Minimize the function z = 3x + 5y subject to the constraints: 
 1. 2y + 5x >= 18 
 2. 6y + 3x >= 27 
 3. y + x >= 6 
 4. x >= 0 
 5. y >= 0

Solve the linear programming problem to minimize z = 3x + 5y with given constraints. Learn how to graph the inequalities, find feasible regions, and calculate corner points to determine the minimum value of z. This problem covers key concepts of linear programming and optimization, suitable for college-level students.

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