Minimize `Z = 3x + 9y` subject to:
`x + 3y le 60, x le y and x, y le 0`.

Answer

To solve the problem of minimizing the objective function \( Z = 3x + 9y \) subject to the given constraints, we will follow these steps:

### Step 1: Identify the Constraints
The constraints given are:
1. \( x + 3y \leq 60 \)
2. \( x \leq y \)
3. \( x \leq 0 \)
4. \( y \leq 0 \)

### Step 2: Convert Inequalities to Equations
To graph the constraints, we convert the inequalities into equations:
1. \( x + 3y = 60 \)
2. \( x = y \)
3. \( x = 0 \)
4. \( y = 0 \)

### Step 3: Find Intercepts
For the first equation \( x + 3y = 60 \):
- When \( x = 0 \), \( 3y = 60 \) → \( y = 20 \) (y-intercept)
- When \( y = 0 \), \( x = 60 \) (x-intercept)

So, the intercepts are \( (60, 0) \) and \( (0, 20) \).

### Step 4: Graph the Constraints
Now, we will graph the constraints:
- The line \( x + 3y = 60 \) will pass through points \( (60, 0) \) and \( (0, 20) \).
- The line \( x = y \) is a diagonal line through the origin with a slope of 1.
- The lines \( x = 0 \) and \( y = 0 \) are the axes.

### Step 5: Determine the Feasible Region
The inequalities indicate that we are interested in the region where:
- Below the line \( x + 3y = 60 \)
- Below the line \( x = y \)
- In the third quadrant (since \( x \leq 0 \) and \( y \leq 0 \))

### Step 6: Identify Vertices of the Feasible Region
The feasible region is bounded by the following points:
- The intersection of \( x + 3y = 60 \) and \( x = y \):
Substitute \( y \) for \( x \) in the first equation:
\[ x + 3x = 60 \implies 4x = 60 \implies x = 15 \implies y = 15 \]
So, one vertex is \( (15, 15) \).

- The point where \( x = 0 \) and \( y = 0 \) is \( (0, 0) \).

### Step 7: Evaluate the Objective Function at Each Vertex
Now we evaluate \( Z = 3x + 9y \) at the vertices:
1. At \( (15, 15) \):
\[ Z = 3(15) + 9(15) = 45 + 135 = 180 \]
2. At \( (0, 0) \):
\[ Z = 3(0) + 9(0) = 0 \]

### Step 8: Determine the Minimum Value
The minimum value of \( Z \) occurs at the vertex \( (0, 0) \) where \( Z = 0 \).

### Conclusion
Thus, the minimum value of \( Z \) is \( 0 \) at the point \( (0, 0) \).

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