
z equals 10 x plus 10 yz=10x+10y

subject to 

7 x plus 9 y7x+9y

greater than or equals≥126126

16 x minus 11 y16x−11y

greater than or equals≥6767

x plus yx+y

less than or equals≤4343

xx

greater than or equals≥0

y

greater than or equals≥0

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**:

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

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 \( y \):

\[
\begin{align*}
77x + 99y &= 1386 \\
144x - 99y &= 603
\end{align*}
\]

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

Learn how to optimize the objective function z = 10x + 10y using linear programming and inequality constraints. This problem involves maximizing z subject to specific linear constraints, suitable for advanced high school students or introductory college courses.

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