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

Options

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

Answer

To find the coordinates of the point at which the minimum value of \( Z = 7x - 8y \) is attained, subject to the constraints given, we will follow these steps:

### Step 1: Identify the Constraints
The constraints are:
1. \( x + y \leq 20 \)
2. \( y \geq 5 \)
3. \( x \geq 0 \)
4. \( y \geq 0 \)

### Step 2: Graph the Constraints
To graph the constraints, we will convert the inequalities into equations and plot them.

1. For \( x + y = 20 \):
- If \( x = 0 \), then \( y = 20 \) (point \( (0, 20) \)).
- If \( y = 0 \), then \( x = 20 \) (point \( (20, 0) \)).
- Draw the line connecting these points.

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

3. For \( x = 0 \) and \( y = 0 \):
- These are the axes of the graph.

### Step 3: Determine the Feasible Region
The feasible region is where all the constraints overlap. This will be bounded by the lines we have drawn and will be in the first quadrant since both \( x \) and \( y \) must be non-negative.

### Step 4: Find the Intersection Points
Next, we need to find the vertices of the feasible region by solving the equations of the lines:

1. Intersection of \( x + y = 20 \) and \( y = 5 \):
- Substitute \( y = 5 \) into \( x + y = 20 \):
\[ x + 5 = 20 \implies x = 15 \]
- So, the intersection point is \( (15, 5) \).

2. Intersection of \( x + y = 20 \) and \( y = 0 \):
- Substitute \( y = 0 \) into \( x + y = 20 \):
\[ x + 0 = 20 \implies x = 20 \]
- So, the intersection point is \( (20, 0) \).

3. Intersection of \( y = 5 \) and \( x = 0 \):
- Substitute \( x = 0 \) into \( y = 5 \):
\[ y = 5 \]
- So, the intersection point is \( (0, 5) \).

### Step 5: Evaluate the Objective Function at Each Vertex
Now we will evaluate \( Z = 7x - 8y \) at each of the vertices found:

1. At \( (15, 5) \):
\[ Z = 7(15) - 8(5) = 105 - 40 = 65 \]

2. At \( (20, 0) \):
\[ Z = 7(20) - 8(0) = 140 - 0 = 140 \]

3. At \( (0, 5) \):
\[ Z = 7(0) - 8(5) = 0 - 40 = -40 \]

### Step 6: Determine the Minimum Value
The values of \( Z \) at the vertices are:
- At \( (15, 5) \): \( Z = 65 \)
- At \( (20, 0) \): \( Z = 140 \)
- At \( (0, 5) \): \( Z = -40 \)

The minimum value of \( Z \) is \( -40 \) at the point \( (0, 5) \).

### Conclusion
Thus, the coordinates of the point at which the minimum value of \( Z = 7x - 8y \) is attained are \( (0, 5) \).