https://file.aiquickdraw.com/mathbot/file/1726395482470voilelkl.jpg

The problem provided is a Linear Programming Problem (LPP) to solve using the Simplex Method. Here's the problem statement based on the image:

### Maximize:
\[ Z = 5x_1 + 4x_2 \]

### Subject to constraints:
1. \( x_1 \leq 7 \)
2. \( x_1 - x_2 \leq 8 \)
3. \( x_1, x_2 \geq 0 \)

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Now, I'll explain the solution steps for solving this LPP using the Simplex Method:

### Step 1: Convert Inequalities to Equalities
To apply the Simplex method, we convert the inequality constraints into equalities by introducing slack variables. Let's introduce slack variables \( s_1 \) and \( s_2 \) for the constraints.

- \( x_1 + s_1 = 7 \)
- \( x_1 - x_2 + s_2 = 8 \)

Here, \( s_1 \) and \( s_2 \geq 0 \) are slack variables.

### Step 2: Set Up the Initial Simplex Table
We rewrite the objective function and constraints in tabular form:

Objective function:
\[ Z = 5x_1 + 4x_2 \]

Thus, the initial tableau is:
| Basis  | \( x_1 \) | \( x_2 \) | \( s_1 \) | \( s_2 \) | RHS  |
|--------|-----------|-----------|-----------|-----------|-------|
| \( s_1 \)  | 1         | 0         | 1         | 0         | 7     |
| \( s_2 \)  | 1         | -1        | 0         | 1         | 8     |
| Z      | -5        | -4        | 0         | 0         | 0     |

### Step 3: Perform Simplex Iterations
We will now perform the Simplex iterations to optimize the objective function by finding the entering and leaving variables, updating the tableau accordingly.

Let me perform the steps and calculate the optimal solution.

I’ll walk through this step and show the solution shortly.The system encountered an issue: the linear program appears to be unbounded. This typically happens when there's no restriction preventing the objective function from increasing infinitely, leading to no optimal solution.

However, let's analyze the problem further. The constraints may allow the objective function \( Z = 5x_1 + 4x_2 \) to increase without bounds, especially considering the directions in which the variables \( x_1 \) and \( x_2 \) can grow.

### Unboundedness Explanation:
- The first constraint is \( x_1 \leq 7 \), which provides a bound for \( x_1 \), but the second constraint \( x_1 - x_2 \leq 8 \) does not restrict \( x_2 \) from growing indefinitely.
- This suggests that increasing \( x_2 \) indefinitely while satisfying the constraints will keep improving the objective function \( Z \), making the problem unbounded.

Would you like to discuss this further, or proceed with adjusting the problem constraints?

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### Relative Questions:
1. What happens when a linear programming problem is unbounded?
2. How do we detect unboundedness in the Simplex method?
3. What are the conditions for boundedness in linear programming problems?
4. Can we modify the constraints to avoid unbounded solutions?
5. What role do slack variables play in the Simplex method?

### Tip:
When solving linear programming problems, always check if the constraints sufficiently limit the variables to prevent unbounded growth.

Solve the given LPP using simplex method. Maximize Z = 5x1 + 4x2 subject to: x1 ≤ 7, x1 - x2 ≤ 8, and x1, x2 ≥ 0.

Learn how to solve a linear programming problem (LPP) using the simplex method. The problem involves maximizing Z = 5x1 + 4x2 subject to constraints x1 ≤ 7, x1 - x2 ≤ 8, and x1, x2 ≥ 0. This tutorial provides step-by-step instructions for setting up and solving the problem using slack variables and the simplex tableau.

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