To solve the problem of minimizing \( Z = 13x - 15y \) subject to the given constraints, we will follow a systematic approach. Here are the steps:
### Step 1: Identify the Constraints
The constraints given are:
1. \( x + y \leq 7 \)
2. \( 2x - 3y + 6 \geq 0 \) or equivalently \( 2x - 3y \geq -6 \)
3. \( x \geq 0 \)
4. \( y \geq 0 \)
### Step 2: Convert Inequalities to Equations
To find the boundary lines of the constraints, we convert the inequalities into equations:
1. \( x + y = 7 \)
2. \( 2x - 3y = -6 \)
### Step 3: Find Intersection Points
Now, we will find the points where these lines intersect the axes and each other.
**For \( x + y = 7 \):**
- If \( x = 0 \), then \( y = 7 \) → Point (0, 7)
- If \( y = 0 \), then \( x = 7 \) → Point (7, 0)
**For \( 2x - 3y = -6 \):**
- If \( x = 0 \), then \( -3y = -6 \) → \( y = 2 \) → Point (0, 2)
- If \( y = 0 \), then \( 2x = -6 \) → \( x = -3 \) (not feasible since \( x \geq 0 \))
### Step 4: Find the Intersection of the Two Lines
To find the intersection of \( x + y = 7 \) and \( 2x - 3y = -6 \):
1. From \( x + y = 7 \), we can express \( y = 7 - x \).
2. Substitute \( y \) into \( 2x - 3y = -6 \):
\[
2x - 3(7 - x) = -6
\]
\[
2x - 21 + 3x = -6
\]
\[
5x - 21 = -6
\]
\[
5x = 15 \implies x = 3
\]
3. Substitute \( x = 3 \) back into \( y = 7 - x \):
\[
y = 7 - 3 = 4
\]
Thus, the intersection point is (3, 4).
### Step 5: Identify the Feasible Region
The feasible region is bounded by the points:
- (0, 0) from \( x \geq 0 \) and \( y \geq 0 \)
- (0, 2) from \( 2x - 3y + 6 \geq 0 \)
- (3, 4) from the intersection of the lines
- (7, 0) from \( x + y = 7 \)
### Step 6: Evaluate the Objective Function at Each Corner Point
Now we evaluate \( Z = 13x - 15y \) at each corner point:
1. At (0, 0):
\[
Z = 13(0) - 15(0) = 0
\]
2. At (0, 2):
\[
Z = 13(0) - 15(2) = -30
\]
3. At (3, 4):
\[
Z = 13(3) - 15(4) = 39 - 60 = -21
\]
4. At (7, 0):
\[
Z = 13(7) - 15(0) = 91
\]
### Step 7: Determine the Minimum Value
The minimum value of \( Z \) occurs at the point (0, 2) where \( Z = -30 \).
### Final Answer
The minimum value of \( Z = 13x - 15y \) is \( -30 \) at the point \( (0, 2) \).
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