 Max Z = 107x1 + x2 + 2x3 
subject to 14x1 + x2 – 6x3 + 3x4≤ 7
 16x1 + 0.5x2 + 6x3 ≤ 5
 3x1 – x2 – x3 ≤ 10

This is a linear programming problem where the objective is to maximize the function \( Z = 107x_1 + x_2 + 2x_3 \), subject to the constraints:

\[
\begin{aligned}
1. & \quad 14x_1 + x_2 - 6x_3 + 3x_4 \leq 7 \\
2. & \quad 16x_1 + 0.5x_2 + 6x_3 \leq 5 \\
3. & \quad 3x_1 - x_2 - x_3 \leq 10 \\
\end{aligned}
\]

To solve this linear programming problem, we can use the **Simplex method**. Here’s the step-by-step approach:

### 1. Convert inequalities into equalities using slack variables
Introduce slack variables \( s_1 \), \( s_2 \), and \( s_3 \) to convert the inequalities to equalities. Slack variables represent the unused resources for each constraint.

\[
\begin{aligned}
1. & \quad 14x_1 + x_2 - 6x_3 + 3x_4 + s_1 = 7 \\
2. & \quad 16x_1 + 0.5x_2 + 6x_3 + s_2 = 5 \\
3. & \quad 3x_1 - x_2 - x_3 + s_3 = 10 \\
\end{aligned}
\]

### 2. Set up the initial simplex tableau
Now, we create the initial simplex tableau by expressing the objective function and the constraints with the slack variables.

| Basis  | \(x_1\) | \(x_2\) | \(x_3\) | \(x_4\) | \(s_1\) | \(s_2\) | \(s_3\) | RHS  |
|--------|---------|---------|---------|---------|---------|---------|---------|------|
| \(s_1\) | 14      | 1       | -6      | 3       | 1       | 0       | 0       | 7    |
| \(s_2\) | 16      | 0.5     | 6       | 0       | 0       | 1       | 0       | 5    |
| \(s_3\) | 3       | -1      | -1      | 0       | 0       | 0       | 1       | 10   |
| \(Z\)   | -107    | -1      | -2      | 0       | 0       | 0       | 0       | 0    |

### 3. Simplex iterations
Perform pivoting and check optimality conditions by selecting entering and leaving variables, and adjusting the tableau iteratively to find the optimal solution.

If you prefer, I can carry out the Simplex method steps and provide the optimal solution. Would you like that?

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Would you like me to go into further details on this?  
Here are 5 related questions to expand on this:

1. What are slack variables and why are they used in linear programming?
2. Can the Simplex method be applied to a minimization problem?
3. How can we identify if a linear programming problem is unbounded?
4. What is the dual of a linear programming problem, and how is it useful?
5. How do we handle integer linear programming problems?

**Tip**: Always check if a linear programming problem is feasible before trying to solve it.

Max Z = 107x1 + x2 + 2x3 subject to 14x1 + x2 – 6x3 + 3x4 ≤ 7, 16x1 + 0.5x2 + 6x3 ≤ 5, 3x1 – x2 – x3 ≤ 10

Learn how to solve a linear programming problem with the objective function Max Z = 107x1 + x2 + 2x3 and three constraints using the Simplex Method. This problem includes the introduction of slack variables and an explanation of how to convert inequalities to equalities.

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