Determine the number of slack variables and name them. Then use the slack variables to convert each constraint into a linear equation. How many and which slack variables should be assigned? OA. There are two slack variables named S₁, S₂. B. There are three slack variables named s₁, $2. $3. OC. There are three slack variables named x₁, x₂. S₁. D. There are five slack variables named x₁, x₂. S₁. Maximize: z=7x₁ + 3x₂ subject to: with 6x₁-x2 s 186 10x₁ +6x₂ ≤ 227 9x₁+x₂336 x₁20, X₂20
Determine the number of slack variables and name them. Then use the slack variables to convert each constraint into a linear equation. How many and which slack variables should be assigned? OA. There are two slack variables named S₁, S₂. B. There are three slack variables named s₁, $2. $3. OC. There are three slack variables named x₁, x₂. S₁. D. There are five slack variables named x₁, x₂. S₁. Maximize: z=7x₁ + 3x₂ subject to: with 6x₁-x2 s 186 10x₁ +6x₂ ≤ 227 9x₁+x₂336 x₁20, X₂20
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![### Linear Programming and Slack Variables
**Determine the Number of Slack Variables and Name Them**
In linear programming, slack variables are added to convert inequalities into equalities in constraints. This assists in solving optimization problems using methods like the Simplex algorithm.
**Problem Statement**
Given the following linear programming problem, determine the number of slack variables and name them. Then use the slack variables to convert each constraint into a linear equation.
#### Objective:
Maximize: \( z = 7x_1 + 3x_2 \)
#### Constraints:
1. \( 6x_1 - x_2 \leq 186 \)
2. \( 10x_1 + 6x_2 \leq 227 \)
3. \( 9x_1 + x_2 \leq 336 \)
Subject to:
\[ x_1 \geq 0, \quad x_2 \geq 0 \]
#### Multiple Choice Question:
How many and which slack variables should be assigned?
- **A.** There are two slack variables named \( s_1, s_2 \).
- **B.** There are three slack variables named \( s_1, s_2, s_3 \).
- **C.** There are three slack variables named \( x_1, x_2, s_1 \).
- **D.** There are five slack variables named \( x_1, x_2, s_1, s_2, s_3 \).
### Explanation
To convert the given inequalities into equalities, we introduce slack variables which represent the difference between the left-hand side and the right-hand side of each constraint. Each constraint will require one slack variable:
1. For \( 6x_1 - x_2 \leq 186 \), we introduce \( s_1 \) such that \( 6x_1 - x_2 + s_1 = 186 \).
2. For \( 10x_1 + 6x_2 \leq 227 \), we introduce \( s_2 \) such that \( 10x_1 + 6x_2 + s_2 = 227 \).
3. For \( 9x_1 + x_2 \leq 336 \), we introduce \( s_3 \) such that \( 9x_1 + x_2 + s_3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F132027c3-2ed1-42c8-912e-c4831680baff%2F319c30de-7643-4195-bd37-12285e03e356%2Fsptvjxm_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Programming and Slack Variables
**Determine the Number of Slack Variables and Name Them**
In linear programming, slack variables are added to convert inequalities into equalities in constraints. This assists in solving optimization problems using methods like the Simplex algorithm.
**Problem Statement**
Given the following linear programming problem, determine the number of slack variables and name them. Then use the slack variables to convert each constraint into a linear equation.
#### Objective:
Maximize: \( z = 7x_1 + 3x_2 \)
#### Constraints:
1. \( 6x_1 - x_2 \leq 186 \)
2. \( 10x_1 + 6x_2 \leq 227 \)
3. \( 9x_1 + x_2 \leq 336 \)
Subject to:
\[ x_1 \geq 0, \quad x_2 \geq 0 \]
#### Multiple Choice Question:
How many and which slack variables should be assigned?
- **A.** There are two slack variables named \( s_1, s_2 \).
- **B.** There are three slack variables named \( s_1, s_2, s_3 \).
- **C.** There are three slack variables named \( x_1, x_2, s_1 \).
- **D.** There are five slack variables named \( x_1, x_2, s_1, s_2, s_3 \).
### Explanation
To convert the given inequalities into equalities, we introduce slack variables which represent the difference between the left-hand side and the right-hand side of each constraint. Each constraint will require one slack variable:
1. For \( 6x_1 - x_2 \leq 186 \), we introduce \( s_1 \) such that \( 6x_1 - x_2 + s_1 = 186 \).
2. For \( 10x_1 + 6x_2 \leq 227 \), we introduce \( s_2 \) such that \( 10x_1 + 6x_2 + s_2 = 227 \).
3. For \( 9x_1 + x_2 \leq 336 \), we introduce \( s_3 \) such that \( 9x_1 + x_2 + s_3
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