`Z = -3x + 4y`
`x + 2y le 8, 3x + 2y le 12, x ge 0, y ge 0`

Answer

To solve the linear programming problem given by the objective function and constraints, we will follow these steps:

### Step 1: Write the Objective Function and Constraints
The objective function we want to maximize or minimize is given by:
\[ Z = -3x + 4y \]

The constraints are:
1. \( x + 2y \leq 8 \)
2. \( 3x + 2y \leq 12 \)
3. \( x \geq 0 \)
4. \( y \geq 0 \)

### Step 2: Convert Constraints to Equations
To find the feasible region, we need to convert the inequalities into equations:
1. \( x + 2y = 8 \)
2. \( 3x + 2y = 12 \)

### Step 3: Find Intercepts of the Constraints
**For the first constraint \( x + 2y = 8 \):**
- **x-intercept:** Set \( y = 0 \):
\[ x + 2(0) = 8 \Rightarrow x = 8 \]
- **y-intercept:** Set \( x = 0 \):
\[ 0 + 2y = 8 \Rightarrow y = 4 \]

So, the intercepts are \( (8, 0) \) and \( (0, 4) \).

**For the second constraint \( 3x + 2y = 12 \):**
- **x-intercept:** Set \( y = 0 \):
\[ 3x + 2(0) = 12 \Rightarrow x = 4 \]
- **y-intercept:** Set \( x = 0 \):
\[ 3(0) + 2y = 12 \Rightarrow y = 6 \]

So, the intercepts are \( (4, 0) \) and \( (0, 6) \).

### Step 4: Plot the Constraints on a Graph
Plot the points found from the intercepts on a graph. The lines will divide the plane into regions. The feasible region will be where all constraints overlap, considering \( x \geq 0 \) and \( y \geq 0 \).

### Step 5: Identify the Feasible Region
The feasible region is bounded by the lines \( x + 2y = 8 \) and \( 3x + 2y = 12 \), along with the axes. The vertices of the feasible region are the points where the constraints intersect.

### Step 6: Find the Intersection of the Constraints
To find the intersection of the lines \( x + 2y = 8 \) and \( 3x + 2y = 12 \):
1. Subtract the first equation from the second:
\[ (3x + 2y) - (x + 2y) = 12 - 8 \]
\[ 2x = 4 \Rightarrow x = 2 \]
2. Substitute \( x = 2 \) into \( x + 2y = 8 \):
\[ 2 + 2y = 8 \Rightarrow 2y = 6 \Rightarrow y = 3 \]

Thus, the intersection point is \( (2, 3) \).

### Step 7: Evaluate the Objective Function at Each Vertex
The vertices of the feasible region are:
1. \( (0, 4) \)
2. \( (2, 3) \)
3. \( (4, 0) \)
4. \( (0, 0) \)

Now, evaluate \( Z = -3x + 4y \) at each vertex:
1. At \( (0, 4) \):
\[ Z = -3(0) + 4(4) = 16 \]
2. At \( (2, 3) \):
\[ Z = -3(2) + 4(3) = -6 + 12 = 6 \]
3. At \( (4, 0) \):
\[ Z = -3(4) + 4(0) = -12 \]
4. At \( (0, 0) \):
\[ Z = -3(0) + 4(0) = 0 \]

### Step 8: Determine Maximum and Minimum Values
From the evaluations:
- Maximum value of \( Z \) is \( 16 \) at \( (0, 4) \).
- Minimum value of \( Z \) is \( -12 \) at \( (4, 0) \).

### Final Answer
- Maximum value of \( Z \) is \( 16 \) at the point \( (0, 4) \).
- Minimum value of \( Z \) is \( -12 \) at the point \( (4, 0) \).

---