Find maximum value of z=2x+3y subject to the constraints `x+y le 4 , x ge 0 , y ge 0` .

Answer

To find the maximum value of \( z = 2x + 3y \) subject to the constraints \( x + y \leq 4 \), \( x \geq 0 \), and \( y \geq 0 \), we can follow these steps:

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

### Step 2: Graph the constraints
To graph the constraint \( x + y = 4 \):
- When \( x = 0 \), \( y = 4 \) (point \( (0, 4) \))
- When \( y = 0 \), \( x = 4 \) (point \( (4, 0) \))

Now, we plot these points on the coordinate plane and draw the line connecting them. The area below this line (including the line itself) represents the region where \( x + y \leq 4 \).

### Step 3: Identify the feasible region
The feasible region is bounded by:
- The line \( x + y = 4 \)
- The x-axis (where \( y = 0 \))
- The y-axis (where \( x = 0 \))

This forms a triangle with vertices at \( (0, 0) \), \( (0, 4) \), and \( (4, 0) \).

### Step 4: Evaluate the objective function at the vertices
We will evaluate \( z = 2x + 3y \) at the vertices of the feasible region:
1. At \( (0, 0) \):
\[ z = 2(0) + 3(0) = 0 \]
2. At \( (0, 4) \):
\[ z = 2(0) + 3(4) = 12 \]
3. At \( (4, 0) \):
\[ z = 2(4) + 3(0) = 8 \]

### Step 5: Determine the maximum value
From the values calculated:
- At \( (0, 0) \), \( z = 0 \)
- At \( (0, 4) \), \( z = 12 \)
- At \( (4, 0) \), \( z = 8 \)

The maximum value of \( z \) occurs at the point \( (0, 4) \) where \( z = 12 \).

### Final Answer
The maximum value of \( z \) is \( \boxed{12} \) at the point \( (0, 4) \).

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