Minimize `Z = x + 4y ` subject to constraints `x + 3y ge 3 , 2x + y ge 2, x ge 0 , y ge 0`

Answer

To solve the problem of minimizing \( Z = x + 4y \) subject to the given constraints, we will follow these steps:

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

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

### Step 3: Find Intercepts of the Lines
For the first equation \( x + 3y = 3 \):
- When \( y = 0 \), \( x = 3 \) (Point: \( (3, 0) \))
- When \( x = 0 \), \( 3y = 3 \) → \( y = 1 \) (Point: \( (0, 1) \))

For the second equation \( 2x + y = 2 \):
- When \( y = 0 \), \( 2x = 2 \) → \( x = 1 \) (Point: \( (1, 0) \))
- When \( x = 0 \), \( y = 2 \) (Point: \( (0, 2) \))

### Step 4: Find the Intersection of the Lines
To find the intersection of the two lines, we solve the equations simultaneously:
1. From \( x + 3y = 3 \), we express \( x \) in terms of \( y \):
\[ x = 3 - 3y \]
2. Substitute \( x \) in the second equation:
\[ 2(3 - 3y) + y = 2 \]
\[ 6 - 6y + y = 2 \]
\[ 6 - 5y = 2 \]
\[ 5y = 4 \quad \Rightarrow \quad y = \frac{4}{5} \]
3. Substitute \( y \) back to find \( x \):
\[ x = 3 - 3\left(\frac{4}{5}\right) = 3 - \frac{12}{5} = \frac{15}{5} - \frac{12}{5} = \frac{3}{5} \]
Thus, the intersection point is \( \left(\frac{3}{5}, \frac{4}{5}\right) \).

### Step 5: Identify the Feasible Region
The feasible region is determined by the inequalities. We plot the lines and shade the appropriate region that satisfies all constraints.

### Step 6: Evaluate the Objective Function at the Corner Points
The corner points of the feasible region are:
1. \( (3, 0) \)
2. \( (0, 2) \)
3. \( \left(\frac{3}{5}, \frac{4}{5}\right) \)

Now we evaluate \( Z = x + 4y \) at each corner point:
1. At \( (3, 0) \):
\[ Z = 3 + 4(0) = 3 \]
2. At \( (0, 2) \):
\[ Z = 0 + 4(2) = 8 \]
3. At \( \left(\frac{3}{5}, \frac{4}{5}\right) \):
\[ Z = \frac{3}{5} + 4\left(\frac{4}{5}\right) = \frac{3}{5} + \frac{16}{5} = \frac{19}{5} = 3.8 \]

### Step 7: Determine the Minimum Value
Comparing the values of \( Z \):
- At \( (3, 0) \), \( Z = 3 \)
- At \( (0, 2) \), \( Z = 8 \)
- At \( \left(\frac{3}{5}, \frac{4}{5}\right) \), \( Z = 3.8 \)

The minimum value of \( Z \) is \( 3 \) at the point \( (3, 0) \).

### Final Answer
The minimum value of \( Z \) is \( 3 \) at the point \( (3, 0) \).

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