Maximize `z =2x + 3y` subject to constraints `x + 4y le 8, 3x + 2y le 14 , x ge 0, y ge 0`.

Answer

To solve the problem of maximizing \( z = 2x + 3y \) subject to the constraints \( x + 4y \leq 8 \), \( 3x + 2y \leq 14 \), \( x \geq 0 \), and \( y \geq 0 \), we will follow these steps:

### Step 1: Identify the Constraints
We have the following constraints:
1. \( x + 4y \leq 8 \)
2. \( 3x + 2y \leq 14 \)
3. \( x \geq 0 \)
4. \( y \geq 0 \)

### Step 2: Convert Inequalities to Equations
To find the boundary lines of the feasible region, we convert the inequalities into equations:
1. \( x + 4y = 8 \)
2. \( 3x + 2y = 14 \)

### Step 3: Find Intercepts of the Constraints
For the first equation \( x + 4y = 8 \):
- When \( x = 0 \), \( 4y = 8 \) → \( y = 2 \) (Point: \( (0, 2) \))
- When \( y = 0 \), \( x = 8 \) (Point: \( (8, 0) \))

For the second equation \( 3x + 2y = 14 \):
- When \( x = 0 \), \( 2y = 14 \) → \( y = 7 \) (Point: \( (0, 7) \))
- When \( y = 0 \), \( 3x = 14 \) → \( x = \frac{14}{3} \approx 4.67 \) (Point: \( \left(\frac{14}{3}, 0\right) \))

### Step 4: Find the Intersection of the Constraints
To find the intersection of the lines \( x + 4y = 8 \) and \( 3x + 2y = 14 \), we can solve these equations simultaneously.

From \( x + 4y = 8 \):
\[ x = 8 - 4y \]

Substituting this into the second equation:
\[ 3(8 - 4y) + 2y = 14 \]
\[ 24 - 12y + 2y = 14 \]
\[ 24 - 10y = 14 \]
\[ -10y = 14 - 24 \]
\[ -10y = -10 \]
\[ y = 1 \]

Now substituting \( y = 1 \) back into \( x + 4y = 8 \):
\[ x + 4(1) = 8 \]
\[ x + 4 = 8 \]
\[ x = 4 \]

So the intersection point is \( (4, 1) \).

### Step 5: Identify the Feasible Region
The feasible region is bounded by the points:
1. \( (0, 2) \)
2. \( (8, 0) \)
3. \( (0, 7) \)
4. \( (4, 1) \)

### Step 6: Evaluate the Objective Function at Each Vertex
Now we evaluate \( z = 2x + 3y \) at each vertex:
1. At \( (0, 2) \): \( z = 2(0) + 3(2) = 6 \)
2. At \( (8, 0) \): \( z = 2(8) + 3(0) = 16 \)
3. At \( (0, 7) \): \( z = 2(0) + 3(7) = 21 \)
4. At \( (4, 1) \): \( z = 2(4) + 3(1) = 8 + 3 = 11 \)

### Step 7: Determine the Maximum Value
The maximum value of \( z \) occurs at \( (8, 0) \) where \( z = 16 \).

### Final Answer
The maximum value of \( z \) is \( 16 \) at the point \( (8, 0) \).

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