Find graphically the minimum value of `Z=5x+7y`, subject to the constraints given below:
`2x+y ge 8, x +2y ge 0 and x, y ge 0`

Answer

To find the minimum value of \( Z = 5x + 7y \) subject to the constraints, we will follow these steps:

### Step 1: Identify the Constraints
The constraints given are:
1. \( 2x + y \geq 8 \)
2. \( x + 2y \geq 0 \)
3. \( x \geq 0 \)
4. \( y \geq 0 \)

### Step 2: Convert Inequalities to Equations
To graph the constraints, we first convert the inequalities into equations:
1. \( 2x + y = 8 \)
2. \( x + 2y = 0 \)

### Step 3: Plot the Constraints
#### For \( 2x + y = 8 \):
- When \( x = 0 \): \( y = 8 \) (Point A: \( (0, 8) \))
- When \( y = 0 \): \( x = 4 \) (Point B: \( (4, 0) \))

#### For \( x + 2y = 0 \):
- When \( x = 0 \): \( y = 0 \) (Point C: \( (0, 0) \))
- When \( y = 0 \): \( x = 0 \) (Point D: \( (0, 0) \))

### Step 4: Determine the Feasible Region
- The line \( 2x + y = 8 \) divides the plane, and we shade the area above this line since we need \( 2x + y \geq 8 \).
- The line \( x + 2y = 0 \) also divides the plane, and we shade the area above this line since we need \( x + 2y \geq 0 \).
- The feasible region is where all shaded areas overlap, which is in the first quadrant due to the constraints \( x \geq 0 \) and \( y \geq 0 \).

### Step 5: Find Intersection Points
To find the vertices of the feasible region, we solve the equations:
1. \( 2x + y = 8 \)
2. \( x + 2y = 0 \)

By substituting \( y = -\frac{x}{2} \) into \( 2x + y = 8 \):
\[ 2x - \frac{x}{2} = 8 \implies \frac{4x - x}{2} = 8 \implies \frac{3x}{2} = 8 \implies x = \frac{16}{3} \]
Substituting \( x = \frac{16}{3} \) back into \( y = -\frac{x}{2} \):
\[ y = -\frac{8}{3} \]
This point is not valid since \( y \) must be non-negative.

### Step 6: Evaluate Corner Points
The corner points of the feasible region are:
1. \( (0, 8) \)
2. \( (4, 0) \)
3. \( (0, 0) \)

Now we evaluate \( Z = 5x + 7y \) at these points:
- At \( (0, 8) \): \( Z = 5(0) + 7(8) = 56 \)
- At \( (4, 0) \): \( Z = 5(4) + 7(0) = 20 \)
- At \( (0, 0) \): \( Z = 5(0) + 7(0) = 0 \)

### Step 7: Determine Minimum Value
The minimum value of \( Z \) occurs at the point \( (4, 0) \) where \( Z = 20 \).

### Conclusion
Thus, the minimum value of \( Z = 5x + 7y \) subject to the given constraints is \( \boxed{20} \).