An objective function and a system of linear inequalities representing constraints are given. Complete parts​ (a) through​ (c).
Objective Function  zequals7 x plus 11 y
Constraints
font size decreased by 1 left brace Start 2 By 1 Matrix 1st Row 1st Column x plus y less than or equals 14 2nd Row 1st Column x plus 2 y less than or equals 18 EndMatrix
 
Question content area bottom left
Part 1
a. Graph the system of inequalities representing the constraints on the given graph of the first Quadrant.
Use the graphing tool to graph the system.

Part 2
b. Find the value of the objective function at each corner of the graphed region bounded by the given constraints and the boundaries of the first​ Quadrant, xgreater than or equals0 and ygreater than or equals0.
The values of the objective function are 
  
enter your response here.
​(Type integers or simplified fractions. Use a comma to separate answers as​ needed.)
Part 3
c. Use the values in part​ (b) to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.
The maximum value of z is zequals
  
enter your response here.
​(Type an integer or a simplified​ fraction.)
Part 4
The maximum occurs at 
  
enter your response here.
​(Type an ordered​ pair.)
.
.
.
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Let's break down the problem and solve it step by step.

### Part 1: Graph the System of Inequalities
The constraints given are:
1. \( x + y \leq 14 \)
2. \( x + 2y \leq 18 \)

Additionally, the problem states the inequalities must be bounded by the first quadrant, so we add:
- \( x \geq 0 \) and \( y \geq 0 \)

To graph the system:
1. **Graph the boundary line** of \( x + y = 14 \):
   - When \( x = 0 \), \( y = 14 \) (point \( (0, 14) \)).
   - When \( y = 0 \), \( x = 14 \) (point \( (14, 0) \)).
   - Draw the line through these points, shading below the line for \( x + y \leq 14 \).

2. **Graph the boundary line** of \( x + 2y = 18 \):
   - When \( x = 0 \), \( 2y = 18 \), so \( y = 9 \) (point \( (0, 9) \)).
   - When \( y = 0 \), \( x = 18 \) (point \( (18, 0) \)).
   - Draw the line through these points, shading below the line for \( x + 2y \leq 18 \).

3. **First Quadrant restrictions**: Shade the area where \( x \geq 0 \) and \( y \geq 0 \), meaning the feasible region is in the first quadrant.

The vertices of the feasible region will be the intersections of these lines and the boundaries.

### Part 2: Find the Objective Function at Each Corner

To solve the objective function \( z = 7x + 11y \), we need the coordinates of the corner points of the feasible region.

The corner points occur at:
1. **Intersection of \( x + y = 14 \) and \( x + 2y = 18 \)**:
   Solve the system of equations:
   \[
   \begin{aligned}
   x + y &= 14 \quad \text{(1)}\\
   x + 2y &= 18 \quad \text{(2)}
   \end{aligned}
   \]
   Subtract equation (1) from equation (2):
   \[
   (x + 2y) - (x + y) = 18 - 14 \implies y = 4
   \]
   Substitute \( y = 4 \) into equation (1):
   \[
   x + 4 = 14 \implies x = 10
   \]
   So, the intersection point is \( (10, 4) \).

2. **Intersection of \( x + y = 14 \) and \( x = 0 \)**:
   When \( x = 0 \), substitute into \( x + y = 14 \):
   \[
   0 + y = 14 \implies y = 14
   \]
   So, this point is \( (0, 14) \).

3. **Intersection of \( x + 2y = 18 \) and \( x = 0 \)**:
   When \( x = 0 \), substitute into \( x + 2y = 18 \):
   \[
   0 + 2y = 18 \implies y = 9
   \]
   So, this point is \( (0, 9) \).

4. **Intersection of \( y = 0 \) and \( x = 0 \)** (Origin):
   The point is \( (0, 0) \).

Now, we have four points: \( (10, 4) \), \( (0, 14) \), \( (0, 9) \), and \( (0, 0) \).

### Part 3: Objective Function Values at Each Corner

Now, plug the corner points into the objective function \( z = 7x + 11y \):

1. At \( (10, 4) \):
   \[
   z = 7(10) + 11(4) = 70 + 44 = 114
   \]

2. At \( (0, 14) \):
   \[
   z = 7(0) + 11(14) = 0 + 154 = 154
   \]

3. At \( (0, 9) \):
   \[
   z = 7(0) + 11(9) = 0 + 99 = 99
   \]

4. At \( (0, 0) \):
   \[
   z = 7(0) + 11(0) = 0
   \]

### Part 4: Maximum Value of the Objective Function

The maximum value of \( z \) occurs at the point \( (0, 14) \), where \( z = 154 \).

### Summary:
- **Maximum value of \( z \)**: \( z = 154 \)
- **Point of maximum \( z \)**: \( (0, 14) \)

Let me know if you'd like more details on any step, or clarification. Here are five questions to further your understanding:

1. How do we interpret the feasible region in the context of linear programming?
2. What is the significance of finding corner points in linear programming problems?
3. Can the maximum or minimum of a linear objective function occur at an interior point of the feasible region?
4. How would the solution change if the constraints were strict inequalities (e.g., \( x + y < 14 \))?
5. How can we extend the idea of objective functions to nonlinear problems?

**Tip:** Always check for corner points of the feasible region when maximizing or minimizing a linear objective function.

An objective function and a system of linear inequalities representing constraints are given. Complete parts (a) through (c). Objective Function: z = 7x + 11y Constraints: x + y ≤ 14, x + 2y ≤ 18.

Learn how to solve a linear programming problem with the objective function z = 7x + 11y and constraints x + y ≤ 14 and x + 2y ≤ 18. Graph the inequalities, find the feasible region, and calculate the maximum value of z at corner points. Suitable for students in Grades 10-12.

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