State True or False:
The maximum value of `Z = 5x + 3y ` subject to constraints `3x + y le 12, 2x + 3y le 18, 0 le x ,y` is 20.

Answer

To determine whether the statement "The maximum value of \( Z = 5x + 3y \) subject to the constraints \( 3x + y \leq 12, 2x + 3y \leq 18, 0 \leq x, y \) is 20" is true or false, we will follow these steps:

### Step 1: Identify the Constraints
The constraints given are:
1. \( 3x + y \leq 12 \)
2. \( 2x + 3y \leq 18 \)
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. \( 3x + y = 12 \)
2. \( 2x + 3y = 18 \)

### Step 3: Find Intercepts of the Lines
For \( 3x + y = 12 \):
- When \( x = 0 \), \( y = 12 \) (point: \( (0, 12) \))
- When \( y = 0 \), \( x = 4 \) (point: \( (4, 0) \))

For \( 2x + 3y = 18 \):
- When \( x = 0 \), \( y = 6 \) (point: \( (0, 6) \))
- When \( y = 0 \), \( x = 9 \) (point: \( (9, 0) \))

### Step 4: Find the Intersection Point
To find the intersection of the lines \( 3x + y = 12 \) and \( 2x + 3y = 18 \), we can solve these equations simultaneously.

From \( 3x + y = 12 \), we can express \( y \):
\[ y = 12 - 3x \]

Substituting \( y \) into the second equation:
\[ 2x + 3(12 - 3x) = 18 \]
\[ 2x + 36 - 9x = 18 \]
\[ -7x + 36 = 18 \]
\[ -7x = 18 - 36 \]
\[ -7x = -18 \]
\[ x = \frac{18}{7} \]

Now substituting \( x \) back to find \( y \):
\[ y = 12 - 3\left(\frac{18}{7}\right) \]
\[ y = 12 - \frac{54}{7} \]
\[ y = \frac{84}{7} - \frac{54}{7} \]
\[ y = \frac{30}{7} \]

So, the intersection point is \( \left(\frac{18}{7}, \frac{30}{7}\right) \).

### Step 5: Identify the Feasible Region
The feasible region is bounded by the points:
- \( (0, 0) \)
- \( (4, 0) \)
- \( (0, 6) \)
- \( \left(\frac{18}{7}, \frac{30}{7}\right) \)

### Step 6: Evaluate \( Z \) at Each Corner Point
Now we will evaluate \( Z = 5x + 3y \) at each corner point:
1. At \( (0, 0) \):
\[ Z = 5(0) + 3(0) = 0 \]
2. At \( (4, 0) \):
\[ Z = 5(4) + 3(0) = 20 \]
3. At \( (0, 6) \):
\[ Z = 5(0) + 3(6) = 18 \]
4. At \( \left(\frac{18}{7}, \frac{30}{7}\right) \):
\[ Z = 5\left(\frac{18}{7}\right) + 3\left(\frac{30}{7}\right) \]
\[ Z = \frac{90}{7} + \frac{90}{7} = \frac{180}{7} \approx 25.71 \]

### Step 7: Determine the Maximum Value
The maximum value of \( Z \) occurs at \( \left(\frac{18}{7}, \frac{30}{7}\right) \) and is approximately \( 25.71 \).

### Conclusion
Since the maximum value of \( Z \) is approximately \( 25.71 \), the statement that the maximum value is 20 is **False**.