To find the point that corresponds with the maximum value of $z = 10x + 10y$ given the constraints:

1. $7x + 9y \geq 126$
2. $16x - 11y \geq 67$
3. $x + y \leq 43$
4. $x \geq 0$
5. $y \geq 0$

We can solve this as a linear programming problem using the method of vertices (corner points). Here's how to proceed:

Step 1: Identify the corner points

We need to find the intersection points (corner points) of the constraints, which will be candidates for the maximum value of $z$.

Step 2: Solve the system of inequalities

1. Intersection of $7x + 9y = 126$ and $16x - 11y = 67$:

   • Solve these two equations simultaneously to find the intersection point.

2. Intersection of $7x + 9y = 126$ and $x + y = 43$:

   • Solve these two equations simultaneously to find the intersection point.

3. Intersection of $16x - 11y = 67$ and $x + y = 43$:

   • Solve these two equations simultaneously to find the intersection point.

4. Check the boundary points where the constraints intersect the $x$- and $y$-axes.

Step 3: Evaluate $z$ at each corner point

Once we have the coordinates of all relevant corner points, we substitute them into the objective function $z = 10x + 10y$ to find the maximum value.

Let's calculate the intersection points and then evaluate $z$ at each one.The intersection points of the constraints are as follows:

1. Intersection of $7x + 9y = 126$ and $16x - 11y = 67$: $(x, y) = (9, 7)$
2. Intersection of $7x + 9y = 126$ and $x + y = 43$: $(x, y) = \left(\frac{261}{2}, \frac{-175}{2}\right)$ (Note: This point is not valid as it has negative $y$, which violates the $y \geq 0$ constraint)
3. Intersection of $16x - 11y = 67$ and $x + y = 43$: $(x, y) = (20, 23)$

Next, we'll evaluate the objective function $z = 10x + 10y$ at the valid corner points to find the maximum $z$.The objective function $z = 10x + 10y$ evaluated at the valid points yields the following:

1. At $(x, y) = (9, 7)$, $z = 160$.
2. At $(x, y) = (20, 23)$, $z = 430$.

Conclusion:

The point $(20, 23)$ corresponds to the maximum value of $z$, which is $430$.

Would you like further details or have any questions?

Additional Questions:

1. How would the solution change if the coefficients in the objective function were different?
2. What happens if we change one of the inequalities, like making $x + y \geq 43$?
3. Can this method be applied if the constraints are non-linear?
4. How do we interpret the results in a real-world scenario?
5. What is the geometric significance of the corner points in linear programming?

Tip:

When solving linear programming problems, always check if all points meet the non-negativity constraints ($x \geq 0$ and $y \geq 0$) before evaluating the objective function.