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

Answer

To solve the linear programming problem of minimizing \( Z = -3x + 4y \) subject to the constraints:

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

we will follow these steps:

### Step 1: Identify the Constraints
We have the following inequalities:
- \( x + 2y \leq 8 \)
- \( 3x + 2y \leq 12 \)
- \( x \geq 0 \)
- \( y \geq 0 \)

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

### Step 3: Find Intercepts
For each equation, we find the x-intercept and y-intercept.

**For \( x + 2y = 8 \):**
- Setting \( y = 0 \): \( x = 8 \) (x-intercept: (8, 0))
- Setting \( x = 0 \): \( 2y = 8 \) → \( y = 4 \) (y-intercept: (0, 4))

**For \( 3x + 2y = 12 \):**
- Setting \( y = 0 \): \( 3x = 12 \) → \( x = 4 \) (x-intercept: (4, 0))
- Setting \( x = 0 \): \( 2y = 12 \) → \( y = 6 \) (y-intercept: (0, 6))

### Step 4: Plot the Lines
Now, we plot the lines on a graph:
- Line 1: Connect points (8, 0) and (0, 4).
- Line 2: Connect points (4, 0) and (0, 6).

### Step 5: Determine Feasible Region
The feasible region is where all the inequalities overlap. Since \( x \geq 0 \) and \( y \geq 0 \), we only consider the first quadrant.

### Step 6: Identify Corner Points
The corner points of the feasible region can be found by solving the equations:
1. Intersection of \( x + 2y = 8 \) and \( 3x + 2y = 12 \):
- Subtract the first from the second:
\[ (3x + 2y) - (x + 2y) = 12 - 8 \implies 2x = 4 \implies x = 2 \]
Substitute \( x = 2 \) into \( x + 2y = 8 \):
\[ 2 + 2y = 8 \implies 2y = 6 \implies y = 3 \]
So, one corner point is \( (2, 3) \).

2. Intersection of \( x + 2y = 8 \) with axes gives points \( (8, 0) \) and \( (0, 4) \).

3. Intersection of \( 3x + 2y = 12 \) with axes gives points \( (4, 0) \) and \( (0, 6) \).

### Step 7: Evaluate the Objective Function at Corner Points
Now we evaluate \( Z = -3x + 4y \) at each corner point:
1. At \( (2, 3) \):
\[ Z = -3(2) + 4(3) = -6 + 12 = 6 \]
2. At \( (8, 0) \):
\[ Z = -3(8) + 4(0) = -24 + 0 = -24 \]
3. At \( (4, 0) \):
\[ Z = -3(4) + 4(0) = -12 + 0 = -12 \]
4. At \( (0, 4) \):
\[ Z = -3(0) + 4(4) = 0 + 16 = 16 \]

### Step 8: Determine Minimum Value
The minimum value of \( Z \) occurs at the point \( (8, 0) \) with \( Z = -24 \).

### Conclusion
The minimum value of \( Z = -3x + 4y \) subject to the given constraints is \( -24 \) at the point \( (8, 0) \).

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