Please show all work when answering the simplex method question. The initial starting point is shown in the image. This is an ungraded lecture problem. ### Problem StatementSolve the following Linear Programming (LP) problem using the simplex method:**Objective Function:**\[\min_{x} \ z = -5x_1 - 7x_2 - 12x_3 + x_4\]**Subject to the Constraints:**1. $ 2x_1 + 3x_2 + 2x_3 + x_4 \leq 38 $2. $ 3x_1 + 2x_2 + 4x_3 - x_4 \leq 55 $**Non-negativity Constraints:**\[ x_1, x_2, x_3, x_4 \geq 0 \]**Initial Solution:**\[ x_{\text{ini}} = (0, 0, 0, 0)^T \]### ExplanationThis problem involves minimizing a linear objective function subject to a set of linear inequalities and non-negativity constraints. The initial solution vector is $(0, 0, 0, 0)$. The simplex method will be applied to find the optimal solution that minimizes the objective function while satisfying all the given constraints.

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Answered: Solve the following LP problem using…

Solve the following LP problem using the simplex method min z = -5x₁ - 7x2 - 12x3 + x4 I s. t. 2x1+3x2+2x3 + x4≤ 38 3x1 + 2x2 + 4x3 − x4 ≤ 55 T1, T2, T3, T4 ≥ 0 Tini = (0,0,0,0)¹.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Please show all work when answering the simplex method question. The initial starting point is shown in the image.

This is an ungraded lecture problem.

### Problem Statement

Solve the following Linear Programming (LP) problem using the simplex method:

**Objective Function:**
\[
\min_{x} \ z = -5x_1 - 7x_2 - 12x_3 + x_4
\]

**Subject to the Constraints:**
1. $ 2x_1 + 3x_2 + 2x_3 + x_4 \leq 38 $
2. $ 3x_1 + 2x_2 + 4x_3 - x_4 \leq 55 $

**Non-negativity Constraints:**
\[ x_1, x_2, x_3, x_4 \geq 0 \]

**Initial Solution:**
\[ x_{\text{ini}} = (0, 0, 0, 0)^T \]

### Explanation

This problem involves minimizing a linear objective function subject to a set of linear inequalities and non-negativity constraints. The initial solution vector is $(0, 0, 0, 0)$. The simplex method will be applied to find the optimal solution that minimizes the objective function while satisfying all the given constraints.

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