### Linear Programming and the Simplex MethodTo solve the given linear programming problem using the Simplex Method, we first need to introduce slack variables to convert the inequalities into equalities and construct the initial simplex tableau. **Problem Statement:**Maximize $ z = x_1 + 9x_2 $**Subject To:**\[\begin{aligned}x_1 + 2x_2 & \leq 12 \\6x_1 + x_2 & \leq 10 \\2x_1 + 2x_2 & \leq 8 \\\end{aligned}\]**Non-negativity Constraints:**\[x_1 \geq 0,\ x_2 \geq 0\]#### Step 1: Introduce Slack VariablesIntroduce slack variables $ s_1 $, $ s_2 $, and $ s_3 $ to convert the inequalities to equalities. The problem now transforms to:\[\begin{aligned}x_1 + 2x_2 + s_1 & = 12 \\6x_1 + x_2 + s_2 & = 10 \\2x_1 + 2x_2 + s_3 & = 8 \\\end{aligned}\]#### Step 2: Construct the Initial Simplex TableauThe initial simplex tableau can be constructed as follows:**Table:**\[\begin{array}{c|cccccc} & x_1 & x_2 & s_1 & s_2 & s_3 & z \\\hline\text{Basic Variable} & & & & & & \\s_1 & 1 & 2 & 1 & 0 & 0 & 12 \\s_2 & 6 & 1 & 0 & 1 & 0 & 10 \\s_3 & 2 & 2 & 0 & 0 & 1 & 8 \\\hlinez & -1 & -9 & 0 & 0 & 0 & 0 \\\end{array}\]### Detailed Explanation of the Initial Simplex Tableau Components- **Row Labels:** "Basic Variable" indicates the current basic variables in the system.- **Columns $x_1$

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Introduce slack variables as necessary and then write the initial simplex tableau for the given linear Maximize z= X₁ + 9x₂ programming problem. subject to X₁ + 2x₂ ≤ 12 6x₁ + x₂ ≤ 10 2x₁ + 2x₂ 58 with x₁ 20, x₂ 20 Complete the initial simplex tableau. x₂ X₂ S₁ $₂ $3 z 1 2 1 0 0 6 0 1 0 0 2 0 0 0 0 0 0 1 10 8 0

Introduce slack variables as necessary and then write the initial simplex tableau for the given linear Maximize z= X₁ + 9x₂ programming problem. subject to X₁ + 2x₂ ≤ 12 6x₁ + x₂ ≤ 10 2x₁ + 2x₂ 58 with x₁ 20, x₂ 20 Complete the initial simplex tableau. x₂ X₂ S₁ $₂ $3 z 1 2 1 0 0 6 0 1 0 0 2 0 0 0 0 0 0 1 10 8 0

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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