To find the maximum value of \( Z = 9x + 13y \) subject to the constraints \( 2x + 3y \leq 18 \), \( 2x + y \leq 10 \), \( x \geq 0 \), and \( y \geq 0 \), we will follow these steps:
### Step 1: Identify the Constraints
The constraints given are:
1. \( 2x + 3y \leq 18 \)
2. \( 2x + y \leq 10 \)
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. \( 2x + 3y = 18 \)
2. \( 2x + y = 10 \)
### Step 3: Find Intercepts of Each Line
For the first line \( 2x + 3y = 18 \):
- When \( x = 0 \): \( 3y = 18 \) → \( y = 6 \) (Point: \( (0, 6) \))
- When \( y = 0 \): \( 2x = 18 \) → \( x = 9 \) (Point: \( (9, 0) \))
For the second line \( 2x + y = 10 \):
- When \( x = 0 \): \( y = 10 \) (Point: \( (0, 10) \))
- When \( y = 0 \): \( 2x = 10 \) → \( x = 5 \) (Point: \( (5, 0) \))
### Step 4: Graph the Constraints
Plot the points \( (0, 6) \), \( (9, 0) \), \( (0, 10) \), and \( (5, 0) \) on a graph. Draw the lines for the equations and shade the feasible region that satisfies all constraints, which is in the first quadrant.
### Step 5: Find the Intersection Points
To find the vertices of the feasible region, we need to find the intersection of the two lines:
1. From \( 2x + 3y = 18 \)
2. From \( 2x + y = 10 \)
We can solve these equations simultaneously:
- From the second equation, express \( y \): \( y = 10 - 2x \)
- Substitute into the first equation:
\[
2x + 3(10 - 2x) = 18
\]
\[
2x + 30 - 6x = 18
\]
\[
-4x + 30 = 18
\]
\[
-4x = -12 \quad \Rightarrow \quad x = 3
\]
- Substitute \( x = 3 \) back into \( y = 10 - 2x \):
\[
y = 10 - 2(3) = 4
\]
- The intersection point is \( (3, 4) \).
### Step 6: Evaluate the Objective Function at Each Vertex
Now we evaluate \( Z = 9x + 13y \) at each vertex of the feasible region:
1. At \( (0, 0) \):
\[
Z = 9(0) + 13(0) = 0
\]
2. At \( (5, 0) \):
\[
Z = 9(5) + 13(0) = 45
\]
3. At \( (0, 6) \):
\[
Z = 9(0) + 13(6) = 78
\]
4. At \( (3, 4) \):
\[
Z = 9(3) + 13(4) = 27 + 52 = 79
\]
### Step 7: Determine the Maximum Value
The maximum value of \( Z \) occurs at the point \( (3, 4) \) and is:
\[
\text{Maximum } Z = 79
\]
### Conclusion
The maximum value of \( Z = 9x + 13y \) subject to the given constraints is **79**.
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