what are the maximum value of z and what points corespond to the maximum value of z

To find the maximum value of \( z = 10x + 10y \) and the corresponding points, we need to evaluate the objective function at the corner points of the feasible region determined by the constraints:

### Constraints Recap:
1. \( 7x + 9y \geq 126 \)
2. \( 16x - 11y \geq 67 \)
3. \( x + y \leq 43 \)
4. \( x \geq 0 \)
5. \( y \geq 0 \)

### Steps:
1. **Graph the constraints** to find the feasible region.
2. **Determine the intersection points** of the constraint lines, which are the corner points of the feasible region.
3. **Evaluate the objective function \( z = 10x + 10y \)** at each corner point.

Let's calculate the intersection points of the constraints analytically and then evaluate the objective function at each point to find the maximum value of \( z \).The feasible corner points and their corresponding values of the objective function \( z = 10x + 10y \) are:

1. Point \( (9, 7) \) with \( z = 160 \).
2. Point \( (20, 23) \) with \( z = 430 \).

### Maximum Value:
- The maximum value of \( z \) is **430** at the point **\( (20, 23) \)**.

Would you like more details on the steps, or do you have any questions?

### Related Questions:
1. How would the optimal solution change if the constraints were modified slightly?
2. What would happen to the feasible region if we added another constraint?
3. How can we interpret this result in a real-world context?
4. What if the objective function changed to \( z = 5x + 15y \)?
5. How can we verify that the feasible region contains all relevant corner points?

**Tip:** Always recheck the constraints at the optimal point to ensure it satisfies all conditions.

Learn how to maximize the objective function z = 10x + 10y within given linear programming constraints. This problem involves evaluating corner points of the feasible region determined by inequalities, and calculating the maximum value of z. Suitable for advanced high school or college-level students studying linear programming.

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