Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**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.](https://content.bartleby.com/qna-images/question/add942ee-f79e-4a36-9f00-66fe52871351/59c2762c-ed25-4265-9d99-f388e9812228/m3r3b4.jpeg)
Transcribed Image Text:**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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