To solve the linear programming problem of maximizing \( Z = 3x + 4y \) subject to the constraints \( x + y \leq 1 \), \( x \geq 0 \), and \( y \geq 0 \), we will follow these steps:
### Step 1: Identify the Constraints
The constraints given are:
1. \( x + y \leq 1 \)
2. \( x \geq 0 \)
3. \( y \geq 0 \)
### Step 2: Convert Inequalities to Equations
To find the boundary lines, we convert the inequalities into equations:
1. \( x + y = 1 \)
2. \( x = 0 \)
3. \( y = 0 \)
### Step 3: Find Intercepts of the Line \( x + y = 1 \)
To plot the line \( x + y = 1 \):
- Set \( x = 0 \):
\[
0 + y = 1 \implies y = 1 \quad \text{(Point: (0, 1))}
\]
- Set \( y = 0 \):
\[
x + 0 = 1 \implies x = 1 \quad \text{(Point: (1, 0))}
\]
### Step 4: Plot the Lines
We will plot the lines on a graph:
- The line \( x + y = 1 \) passes through the points (0, 1) and (1, 0).
- The line \( x = 0 \) is the y-axis.
- The line \( y = 0 \) is the x-axis.
### Step 5: Identify the Feasible Region
The feasible region is where all constraints overlap. Since \( x + y \leq 1 \), the feasible region will be below the line \( x + y = 1 \) and in the first quadrant (where \( x \geq 0 \) and \( y \geq 0 \)). The feasible region is bounded by the points:
- (0, 0)
- (1, 0)
- (0, 1)
### Step 6: Evaluate the Objective Function at the Corner Points
We will evaluate \( Z = 3x + 4y \) at each corner point:
1. At (0, 0):
\[
Z = 3(0) + 4(0) = 0
\]
2. At (1, 0):
\[
Z = 3(1) + 4(0) = 3
\]
3. At (0, 1):
\[
Z = 3(0) + 4(1) = 4
\]
### Step 7: Determine the Maximum Value
Comparing the values of \( Z \):
- At (0, 0), \( Z = 0 \)
- At (1, 0), \( Z = 3 \)
- At (0, 1), \( Z = 4 \)
The maximum value of \( Z \) is \( 4 \) at the point (0, 1).
### Final Answer
The maximum value of \( Z \) is \( 4 \) at the point \( (0, 1) \).
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