https://file.aiquickdraw.com/mathbot/file/1733125805404h8l22yi9.jpg

To solve the linear programming problem in the uploaded image, let’s break it down step by step based on the problem's requirements:

### Problem Overview:
Objective: Minimize \( Z = 3x + 12y \)  
Subject to constraints:
1. \( 5x + y \leq 32 \)
2. \( x + 3y \geq 12 \)
3. \( x, y \geq 0 \) (non-negativity constraints)

---

### Steps:
#### 1. **Find the x- and y-intercepts for each constraint**:
   - For \( 5x + y = 32 \):
     - Set \( x = 0 \): \( y = 32 \).
     - Set \( y = 0 \): \( x = 6.4 \) (\( 32/5 \)).

   - For \( x + 3y = 12 \):
     - Set \( x = 0 \): \( y = 4 \).
     - Set \( y = 0 \): \( x = 12 \).

#### 2. **Graph the constraints**:
   - Plot the lines for \( 5x + y = 32 \) and \( x + 3y = 12 \) using their intercepts.
   - Shade the feasible region that satisfies both constraints:
     - Below \( 5x + y = 32 \) (due to \( \leq \)).
     - Above \( x + 3y = 12 \) (due to \( \geq \)).
     - Ensure the region is within the first quadrant (non-negativity).

#### 3. **Point of Intersection**:
   - Solve the equations \( 5x + y = 32 \) and \( x + 3y = 12 \) simultaneously:
     - From \( 5x + y = 32 \), express \( y \): \( y = 32 - 5x \).
     - Substitute \( y = 32 - 5x \) into \( x + 3y = 12 \):
       \[
       x + 3(32 - 5x) = 12
       \]
       \[
       x + 96 - 15x = 12
       \]
       \[
       -14x + 96 = 12
       \]
       \[
       14x = 84 \implies x = 6
       \]
     - Substitute \( x = 6 \) into \( y = 32 - 5x \):
       \[
       y = 32 - 5(6) = 32 - 30 = 2
       \]
     - Intersection point: \( (6, 2) \).

#### 4. **Corner Points**:
   - Corner points of the feasible region:
     1. \( (0, 4) \) (intercept of \( x + 3y = 12 \)).
     2. \( (6, 2) \) (intersection of \( 5x + y = 32 \) and \( x + 3y = 12 \)).
     3. \( (6.4, 0) \) (intercept of \( 5x + y = 32 \)).

#### 5. **Substitute into the Objective Function**:
   - Evaluate \( Z = 3x + 12y \) for each corner point:
     - At \( (0, 4) \): \( Z = 3(0) + 12(4) = 48 \).
     - At \( (6, 2) \): \( Z = 3(6) + 12(2) = 18 + 24 = 42 \).
     - At \( (6.4, 0) \): \( Z = 3(6.4) + 12(0) = 19.2 \).

#### 6. **Formulate the Decision**:
   - Minimum \( Z \) occurs at \( (6.4, 0) \) with \( Z = 19.2 \).
   - Decision:
     - \( x = 6.4 \),
     - \( y = 0 \),
     - \( \text{Minimum } Z = 19.2 \).

---

Would you like a detailed graph or further explanation on any step?  
### Related Questions:
1. How can you graph the feasible region manually?
2. Why does the intersection point help solve the problem?
3. How would the solution change if the objective was to maximize \( Z \)?
4. Can we use another method like the Simplex Method to solve this?
5. What are practical scenarios where linear programming is used?

### Tip:
Always confirm that your solution is in the feasible region by substituting back into all constraints.

Using the graphical method, find the values of x and y to solve the linear programming problem: Minimize Z = 3x + 12y, subject to the constraints 5x + y ≤ 32, x + 3y ≥ 12, and x, y ≥ 0.

Learn how to solve a linear programming problem using the graphical method. Minimize Z = 3x + 12y subject to constraints like 5x + y ≤ 32, x + 3y ≥ 12, and non-negativity conditions. This solution explores feasible regions, corner points, and optimization techniques.

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