Maximize `Z = 4x + y` subject to : `x + y le 30, x, y ge 0`.

Answer

To solve the problem of maximizing \( Z = 4x + y \) subject to the constraints \( x + y \leq 30 \) and \( x, y \geq 0 \), we will follow these steps:

### Step 1: Identify the Constraints
The constraints given are:
1. \( x + y \leq 30 \)
2. \( x \geq 0 \)
3. \( y \geq 0 \)

### Step 2: Convert the Inequality to an Equation
To find the boundary of the feasible region, we convert the inequality \( x + y \leq 30 \) into an equation:
\[ x + y = 30 \]

### Step 3: Determine the Intercepts
To graph the line \( x + y = 30 \), we find the intercepts:
- **X-intercept**: Set \( y = 0 \):
\[ x + 0 = 30 \Rightarrow x = 30 \]
So, the x-intercept is \( (30, 0) \).

- **Y-intercept**: Set \( x = 0 \):
\[ 0 + y = 30 \Rightarrow y = 30 \]
So, the y-intercept is \( (0, 30) \).

### Step 4: Graph the Constraints
Now we can plot the line \( x + y = 30 \) on the graph. The line will pass through the points \( (30, 0) \) and \( (0, 30) \).

### Step 5: Shade the Feasible Region
Since we have the inequality \( x + y \leq 30 \), we shade the area below the line. Additionally, since \( x \geq 0 \) and \( y \geq 0 \), we only consider the first quadrant where both \( x \) and \( y \) are non-negative.

### Step 6: Identify the Corner Points
The feasible region is bounded by the axes and the line \( x + y = 30 \). The corner points of this region are:
1. \( (0, 0) \)
2. \( (30, 0) \)
3. \( (0, 30) \)

### Step 7: Evaluate the Objective Function at Each Corner Point
Now, we will evaluate \( Z = 4x + y \) at each of the corner points:
- At \( (0, 0) \):
\[ Z = 4(0) + 0 = 0 \]

- At \( (30, 0) \):
\[ Z = 4(30) + 0 = 120 \]

- At \( (0, 30) \):
\[ Z = 4(0) + 30 = 30 \]

### Step 8: Determine the Maximum Value
From the evaluations:
- \( Z(0, 0) = 0 \)
- \( Z(30, 0) = 120 \)
- \( Z(0, 30) = 30 \)

The maximum value of \( Z \) occurs at the point \( (30, 0) \) and is:
\[ \text{Maximum } Z = 120 \]

### Conclusion
Thus, the maximum value of \( Z = 4x + y \) subject to the given constraints is \( 120 \) at the point \( (30, 0) \).

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