To solve this problem, let's break down the given information and constraints step-by-step.

You have an optimization problem where you need to maximize or minimize the function $z = 10x + 10y$ subject to certain constraints. Here is the formal statement of the problem:

Objective Function

$z = 10x + 10y$

Constraints

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

Steps to Solve the Problem:

1. Graph the Constraints: Each inequality constraint represents a region on a coordinate plane. The feasible region, which satisfies all constraints, is the intersection of these regions.

2. Find the Intersection Points (Vertices): Identify the points where the constraint lines intersect. These points are potential candidates for maximizing or minimizing the objective function.

3. Evaluate the Objective Function at Each Vertex: Compute the value of $z = 10x + 10y$ at each vertex to determine which point gives the maximum or minimum value, depending on the goal.

Step-by-Step Solution:

1. Rewrite Constraints for Graphing:

   7x + 9y &\geq 126 \quad \text{(Constraint 1)} \\ 16x - 11y &\geq 67 \quad \text{(Constraint 2)} \\ x + y &\leq 43 \quad \text{(Constraint 3)} \\ x &\geq 0 \quad \text{(Constraint 4)} \\ y &\geq 0 \quad \text{(Constraint 5)} \end{align*}$$

2. Find Intersections of Lines:

   • Solve for $y$ in terms of $x$ for each constraint to get lines in slope-intercept form.
   • Find intersections by setting equations equal to each other.

3. Solve for Vertices:

Let's solve the system step-by-step:

Constraint 1 and Constraint 2 Intersection:

$7x + 9y = 126 \quad \text{and} \quad 16x - 11y = 67$

Multiply the first equation by 11 and the second by 9 to eliminate $y$:

77x + 99y &= 1386 \\ 144x - 99y &= 603 \end{align*}$$ Add the two equations: $$221x = 1989 \implies x = \frac{1989}{221} \approx 9$$ Substitute $$x = 9$$ back into one of the original equations to find $$y$$: $$7(9) + 9y = 126 \implies 63 + 9y = 126 \implies 9y = 63 \implies y = 7$$ So, one vertex is $$(9, 7)$$. #### Constraint 1 and Constraint 3 Intersection: $$7x + 9y = 126 \quad \text{and} \quad x + y = 43$$ Substitute $$y = 43 - x$$ into the first equation: $$7x + 9(43 - x) = 126 \implies 7x + 387 - 9x = 126 \implies -2x = -261 \implies x = 130.5 \text{ (Not feasible)}$$ We discard the non-feasible solution. #### Constraint 2 and Constraint 3 Intersection: $$16x - 11y = 67 \quad \text{and} \quad x + y = 43$$ Substitute $$y = 43 - x$$ into the second equation: $$16x - 11(43 - x) = 67 \implies 16x - 473 + 11x = 67 \implies 27x = 540 \implies x = 20$$ Then find $$y$$: $$y = 43 - 20 = 23$$ So another vertex is $$(20, 23)$$. 4. **Evaluate Objective Function at Each Vertex**: - At $$(9, 7)$$, $$z = 10(9) + 10(7) = 90 + 70 = 160$$. - At $$(20, 23)$$, $$z = 10(20) + 10(23) = 200 + 230 = 430$$. ### Conclusion: - The maximum value of $$z$$ is **430** at the point $$(20, 23)$$. Would you like more details or have any questions about this solution? ### Follow-Up Questions: 1. What are the characteristics of the feasible region for this linear programming problem? 2. How would the solution change if the objective function was to minimize $$z = 10x + 10y$$ instead? 3. Can the solution be solved graphically, and if so, how? 4. What impact do the non-negativity constraints ($$x \geq 0$$, $$y \geq 0$$) have on the solution? 5. How can we verify