Maximize and minimize z=3x+4y and 2x+y≥10 and x+2y≥8 and x, y≥0

To maximize and minimize \( z = 3x + 4y \) subject to the constraints:

1. \( 2x + y \geq 10 \)
2. \( x + 2y \geq 8 \)
3. \( x, y \geq 0 \)

### Step 1: Identify the Feasible Region

1. **Rewrite the inequalities** as equalities to find boundary lines:
   - For \( 2x + y = 10 \):
     - When \( x = 0 \): \( y = 10 \)
     - When \( y = 0 \): \( x = 5 \)
     - Points: \( (0, 10) \) and \( (5, 0) \)
   - For \( x + 2y = 8 \):
     - When \( x = 0 \): \( y = 4 \)
     - When \( y = 0 \): \( x = 8 \)
     - Points: \( (0, 4) \) and \( (8, 0) \)

2. **Plot the lines** and determine the feasible region, which is the area where both inequalities \( 2x + y \geq 10 \) and \( x + 2y \geq 8 \) are satisfied along with \( x, y \geq 0 \).

3. **Find Intersection Points** of the boundary lines:
   - Solve \( 2x + y = 10 \) and \( x + 2y = 8 \) simultaneously:
     - From \( 2x + y = 10 \), we get \( y = 10 - 2x \).
     - Substitute into \( x + 2(10 - 2x) = 8 \):
       \[
       x + 20 - 4x = 8 \Rightarrow -3x = -12 \Rightarrow x = 4
       \]
       \[
       y = 10 - 2(4) = 2
       \]
     - Intersection point: \( (4, 2) \)

4. **Vertices of the feasible region** are \( (0, 10) \), \( (5, 0) \), \( (8, 0) \), and \( (4, 2) \).

### Step 2: Evaluate \( z = 3x + 4y \) at Each Vertex

1. At \( (0, 10) \):
   \[
   z = 3(0) + 4(10) = 40
   \]

2. At \( (5, 0) \):
   \[
   z = 3(5) + 4(0) = 15
   \]

3. At \( (8, 0) \):
   \[
   z = 3(8) + 4(0) = 24
   \]

4. At \( (4, 2) \):
   \[
   z = 3(4) + 4(2) = 12 + 8 = 20
   \]

### Step 3: Conclusion

- **Maximum value** of \( z = 40 \) at \( (0, 10) \).
- **Minimum value** of \( z = 15 \) at \( (5, 0) \).

Would you like a more detailed explanation or have any questions?

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Here are some additional questions for practice:
1. How would the solution change if \( z = 5x + 3y \)?
2. What would happen if we changed one of the constraints, such as \( 2x + y \leq 10 \)?
3. How do we determine if a solution is bounded or unbounded?
4. How would you handle a constraint like \( y \leq 6 \) in this problem?
5. What if we were asked to only maximize (or only minimize) \( z \)?

**Tip:** When working with linear programming, always verify the feasibility of intersection points within the constraints.

Explore how to maximize and minimize the function z = 3x + 4y subject to constraints 2x + y ≥ 10 and x + 2y ≥ 8, using linear programming techniques. This solution includes steps to determine the feasible region and evaluate the objective function at vertices, ideal for Grades 10-12 students.

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