**Linear Programming Problem with Slack Variables**In this exercise, you'll learn to transform a linear programming problem using slack variables. The objective is to convert inequalities into equations to solve the problem optimally.**Objective:**Maximize the function:\[ P = 3x_1 + 5x_2 \]**Constraints:**1. $ 8x_1 + 2x_2 \leq 15 $2. $ 6x_1 + 5x_2 \leq 12 $3. Non-negativity: $ x_1, x_2 \geq 0 $**Task:**Convert the constraint $ 8x_1 + 2x_2 \leq 15 $ into an equation by introducing a slack variable $ s_1 $.**Solution:**Rewrite the inequality as an equation:\[ 8x_1 + 2x_2 + s_1 = 15 \]Ensure that $ s_1 \geq 0 $ as it represents the unused resources or 'slack' in the system.By transforming the inequalities into equations, you can analyze and solve the linear programming problem using methods such as the Simplex Algorithm.

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Answered: Write the e-system obtained via slack…

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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Write the e-system obtained via slack variables for the given linear programming problem. Maximize P = 3x, + 5x2 subject to 8x, +2x2 s 15 6x, + 5x2 s 12 X4, X2 2 0