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The image contains three linear programming problems that ask to minimize a given objective function, subject to a set of constraints. Let's break each of them down.

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### Problem 1:

\[
f = 2x_1 - x_2 - x_4 \to \min
\]

**Subject to:**

\[
\begin{aligned}
x_1 - 2x_2 + x_3 &= 10, \\
-2x_1 - x_2 - 2x_4 &\geq 18, \\
3x_1 + 2x_2 + x_4 &\geq 36, \\
x_j &\geq 0, \quad j = 1, 4.
\end{aligned}
\]

---

### Problem 2:

\[
f = 2x_1 - x_2 - x_4 \to \min
\]

**Subject to:**

\[
\begin{aligned}
x_1 - 2x_2 + x_3 &= 10, \\
-2x_1 - x_2 - 2x_4 - x_5 &= 18, \\
3x_1 + 2x_2 + x_4 - x_6 &= 36, \\
x_j &\geq 0, \quad j = 1, 6.
\end{aligned}
\]

---

### Problem 3:

\[
f = 2x_1 - x_2 - x_4 + Mx_7 + Mx_8 \to \min
\]

**Subject to:**

\[
\begin{aligned}
x_1 - 2x_2 + x_3 &= 10, \\
-2x_1 - x_2 - 2x_4 - x_5 + x_7 &= 18, \\
3x_1 + 2x_2 + x_4 - x_6 + x_8 &= 36, \\
x_j &\geq 0, \quad j = 1, 8.
\end{aligned}
\]

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### Summary:

- All problems minimize the function \( f = 2x_1 - x_2 - x_4 \), with additional terms involving \( Mx_7 \) and \( Mx_8 \) in Problem 3.
- The constraints are a system of linear equations and inequalities with non-negativity conditions for the variables \( x_j \), where \( j \) ranges over the problem's specific variables.

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Would you like to see the solutions for these problems, or further analysis on a specific one?

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Here are five questions to consider:
1. What is the method for solving such linear programming problems?
2. How can the Simplex Method be applied to these problems?
3. What role do the slack variables play in converting inequalities to equalities?
4. How can duality theory be used to analyze these linear programs?
5. What is the significance of adding \( Mx_7 \) and \( Mx_8 \) in the third problem?

**Tip:** In linear programming, using the Simplex method often requires converting inequalities into equalities by introducing slack or surplus variables to handle constraints efficiently.

The image contains three linear programming problems that ask to minimize a given objective function, subject to a set of constraints. The problems are represented mathematically.

Explore the step-by-step solution of linear programming problems that involve minimizing an objective function with multiple constraints. These problems include concepts such as slack variables, simplex method, and duality theory. Suitable for undergraduate students or advanced high school learners.

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