Question

linear programming problem solution 2x+4y ≤ 5, 9x + 8y ≥ 5 and 3x+2y ≤ 5

Answer 1

To solve the given linear programming problem, we have the following constraints:

1. \( 2x + 4y \leq 5 \) (Constraint 1)
2. \( 9x + 8y \geq 5 \) (Constraint 2)
3. \( 3x + 2y \leq 5 \) (Constraint 3)

We want to determine the feasible region defined by these inequalities and then optimize a function (usually a maximization or minimization of an objective function). However, you have not mentioned an objective function to optimize. For demonstration purposes, let's assume we want to maximize \( z = x + y \).

### Step 1: Graph the Constraints

We'll convert each inequality into an equation to find the boundary lines:

1. **For \( 2x + 4y = 5 \)**:
- When \( x = 0 \): \( 4y = 5 \Rightarrow y = \frac{5}{4} = 1.25 \)
- When \( y = 0 \): \( 2x = 5 \Rightarrow x = \frac{5}{2} = 2.5 \)
- Points: \( (0, 1.25) \) and \( (2.5, 0) \)

2. **For \( 9x + 8y = 5 \)**:
- When \( x = 0 \): \( 8y = 5 \Rightarrow y = \frac{5}{8} = 0.625 \)
- When \( y = 0 \): \( 9x = 5 \Rightarrow x = \frac{5}{9} \approx 0.555 \)
- Points: \( (0, 0.625) \) and \( (0.555, 0) \)

3. **For \( 3x + 2y = 5 \)**:
- When \( x = 0 \): \( 2y = 5 \Rightarrow y = \frac{5}{2} = 2.5 \)
- When \( y = 0 \): \( 3x = 5 \Rightarrow x = \frac{5}{3} \approx 1.667 \)
- Points: \( (0, 2.5) \) and \( (1.667, 0) \)

### Step 2: Plot the Feasible Region

1. **Identify the Inequalities**:
- The region for \( 2x + 4y \leq 5 \) is below the line.
- The region for \( 9x + 8y \geq 5 \) is above the line.
- The region for \( 3x + 2y \leq 5 \) is below the line.

2. **Draw the lines** and shade appropriate regions.

### Step 3: Find the Corner Points

To find the feasible region's vertices, solve the system of equations formed by pairs of constraints (only consider intersections of constraints that apply).

1. **Intersection of the first and third equations**:
\[
2x + 4y = 5
\]
\[
3x + 2y = 5
\]
Multiply the second equation by 2:
\[
6x + 4y = 10
\]
Subtract the first from the modified second:
\[
4x = 5 \Rightarrow x = \frac{5}{4} = 1.25
\]
Substitute \( x \) back into first equation to find \( y \):
\[
2(1.25) + 4y = 5 \Rightarrow 2.5 + 4y = 5 \Rightarrow 4y = 2.5 \Rightarrow y = 0.625
\]
Vertex: \( (1.25, 0.625) \)

2. **Intersection of first and second equations**:
\[
2x + 4y = 5
\]
\[
9x + 8y = 5
\]
Multiply the first equation by 2:
\[
4x + 8y = 10
\]
Subtract:
\[
5x = 5 \Rightarrow x = 1
\]
Substitute back into the first equation:
\[
2(1) + 4y = 5 \Rightarrow 2 + 4y = 5 \Rightarrow 4y = 3 \Rightarrow y = \frac{3}{4} = 0.75
\]
Vertex: \( (1, 0.75) \)

3. **Intersection of the second and third equations**:
\[
9x + 8y = 5
\]
\[
3x + 2y = 5
\]
Follow similar substitution and elimination methods to find this intersection point.

### Step 4: Evaluate the Objective Function at Each Vertex

Calculate \( z = x + y \) for each vertex found:
- At \( (1.25, 0.625) \): \( z = 1.25 + 0.625 = 1.875 \)
- At \( (1, 0.75) \): \( z = 1 + 0.75 = 1.75 \)
- Evaluate last intersection similarly.

### Step 5: Identify Optimal Solution

Compare the values of \( z \) calculated and choose the maximum:

In this case, if \( (1.25, 0.625) \) gives the highest value, that would be your solution based on the assumed objective function \( z = x + y \).

### Conclusion

To find the feasibility region and vertices accurately, graphical tools or software will provide a clearer view. The provided outlines should lead you to the solution or an optimal point based on the situation.