The coordinate of the point at which minimum value of Z = 7x - 8y, subject to the conditions constraints `x+y - 20 le 0, y ge 5, le -5 ` is

Options

A. (20,0)
B. (15,5)
C. (0,5)
D. (0,20)

Explanation

To find the coordinate of the point at which the minimum value of \( Z = 7x - 8y \) occurs, subject to the constraints \( x + y \leq 20 \), \( y \geq 5 \), and \( x \leq -5 \), we can follow these steps:

### Step 1: Identify the Constraints
The constraints given are:
1. \( x + y \leq 20 \)
2. \( y \geq 5 \)
3. \( x \leq -5 \)

### Step 2: Graph the Constraints
To graph the constraints, we can convert them into equations:
- For \( x + y = 20 \):
- When \( x = 0 \), \( y = 20 \) (point \( (0, 20) \))
- When \( y = 0 \), \( x = 20 \) (point \( (20, 0) \))

- For \( y = 5 \):
- This is a horizontal line at \( y = 5 \).

- For \( x = -5 \):
- This is a vertical line at \( x = -5 \).

### Step 3: Determine the Feasible Region
The feasible region is the area that satisfies all the constraints. It is bounded by the lines we have drawn:
- The line \( x + y = 20 \) will intersect the line \( y = 5 \) at the point where \( y = 5 \):
\[ x + 5 = 20 \implies x = 15 \quad \text{(Point (15, 5))} \]
- The intersection of \( y = 5 \) and \( x = -5 \) gives the point \( (-5, 5) \).

### Step 4: Identify the Vertices of the Feasible Region
The vertices of the feasible region formed by the constraints are:
1. \( (15, 5) \)
2. \( (-5, 5) \)
3. \( (-5, 20) \) (not valid since it exceeds \( y = 5 \))

### Step 5: Evaluate the Objective Function at Each Vertex
Now we will evaluate \( Z = 7x - 8y \) at the vertices:
1. At \( (15, 5) \):
\[ Z = 7(15) - 8(5) = 105 - 40 = 65 \]
2. At \( (-5, 5) \):
\[ Z = 7(-5) - 8(5) = -35 - 40 = -75 \]

### Step 6: Determine the Minimum Value
From the evaluations:
- \( Z(15, 5) = 65 \)
- \( Z(-5, 5) = -75 \)

The minimum value occurs at the point \( (-5, 5) \).

### Final Answer
The coordinate of the point at which the minimum value of \( Z \) occurs is:
\[ \boxed{(-5, 5)} \]