Maximize z = -x1 +2x2 -x3 
Subject to constraints:
3x1 + x2 -x3  less than or equal to 10
-x1 +4x2 +x3  greater than or equal to 6
x2 + x3 less than or equal to 4

Let's solve the linear programming problem you've provided:

### Problem Statement:
**Maximize:**  
\[ z = -x_1 + 2x_2 - x_3 \]

**Subject to the constraints:**  
1. \( 3x_1 + x_2 - x_3 \leq 10 \)
2. \( -x_1 + 4x_2 + x_3 \geq 6 \)
3. \( x_2 + x_3 \leq 4 \)
4. \( x_1, x_2, x_3 \geq 0 \) (Assuming non-negativity unless specified otherwise)

### Step 1: Convert Inequalities to Standard Form
Before solving the problem using the simplex method, let's convert all the inequalities into a standard form (with all inequalities of the "less than or equal to" type).

1. \( 3x_1 + x_2 - x_3 \leq 10 \) (Already in standard form)
2. \( -x_1 + 4x_2 + x_3 \geq 6 \)  
   This can be rewritten as:  
   \( x_1 - 4x_2 - x_3 \leq -6 \)
3. \( x_2 + x_3 \leq 4 \) (Already in standard form)

### Step 2: Introduce Slack Variables
We introduce slack variables to convert the inequalities into equalities.

1. \( 3x_1 + x_2 - x_3 + s_1 = 10 \) where \( s_1 \geq 0 \).
2. \( x_1 - 4x_2 - x_3 + s_2 = -6 \) where \( s_2 \geq 0 \).
3. \( x_2 + x_3 + s_3 = 4 \) where \( s_3 \geq 0 \).

### Step 3: Set Up the Initial Simplex Tableau
We can now set up the initial simplex tableau.

### Step 4: Perform Simplex Iterations
Let's perform the necessary simplex iterations to find the optimal solution.### Solution of the Linear Programming Problem:

The optimization was successful, and the maximum value of \( z \) is:

\[ z = -2 \]

### Optimal Solution:
The values of the variables at the optimal solution are:
- \( x_1 = 3.45 \) (approximately)
- \( x_2 = 1.82 \) (approximately)
- \( x_3 = 2.18 \) (approximately)

### Interpretation:
At this optimal solution, the objective function achieves a maximum value of \( z = -2 \), given the constraints provided.

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Would you like a more detailed explanation of the steps involved or have any specific questions?

Here are 8 relative questions that you might be interested in:

1. How do slack variables affect the solution in linear programming?
2. What is the significance of the dual problem in linear programming?
3. How would the solution change if one of the constraints was altered?
4. What are the differences between the simplex and dual simplex methods?
5. How does the choice of method (e.g., simplex vs. interior-point) affect the result?
6. Can linear programming be applied to non-linear problems?
7. What are the practical applications of the solution found in this linear program?
8. How would we interpret the results if the problem was a minimization instead of maximization?

**Tip:** When dealing with linear programming problems, always convert inequalities to standard form and introduce slack variables to simplify the application of the simplex method.

Learn how to maximize the objective function z = -x1 + 2x2 - x3 using linear programming and the simplex method. Explore the detailed solution involving inequalities and optimal variable values. Suitable for college-level students and professionals.

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