Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Systems of Equations and Basic Solutions
The following section provides educational content on solving a system of linear equations using the concept of basic solutions.
#### System of Equations:
The given system of linear equations is:
\[
\begin{align*}
1. & \quad x_1 + x_2 + s_1 = 25 \\
2. & \quad 3x_1 + x_2 + s_2 = 33 \\
3. & \quad 5x_1 + x_2 + s_3 = 45 \\
\end{align*}
\]
In this system of equations, \(x_1\) and \(x_2\) are the variables, while \(s_1\), \(s_2\), and \(s_3\) are the slack variables.
#### Basic Solutions:
A partially completed table is provided below showing ten solutions to the system of equations. Solutions (A) to (J) consist of various values assigned to \(x_1\), \(x_2\), \(s_1\), \(s_2\), and \(s_3\):
| Solutions | \(x_1\) | \(x_2\) | \(s_1\) | \(s_2\) | \(s_3\) |
|-----------|-----|-----|-----|-----|-----|
| (A) | 0 | 0 | 25 | 33 | 45 |
| (B) | 0 | 25 | 0 | 8 | 20 |
| (C) | 0 | 33 | -8 | 0 | 12 |
| (D) | 0 | 45 | -20 | -12 | 0 |
| (E) | 25 | 0 | 0 | -42 | -80 |
| (F) | 11 | 0 | 14 | -10 | 0 |
| (G) | 0 | 0 | 0 | 0 | 0 |
| (H) | 0 | 0 | 0 | 0 | 0 |
| (I) |](https://content.bartleby.com/qna-images/question/ebf1d700-a6c3-4dc9-8bae-c4d04bafe1e4/d0da254b-a952-4ae2-9cc0-abd7e569add7/wfsmxc6_processed.jpeg)
Transcribed Image Text:### Systems of Equations and Basic Solutions
The following section provides educational content on solving a system of linear equations using the concept of basic solutions.
#### System of Equations:
The given system of linear equations is:
\[
\begin{align*}
1. & \quad x_1 + x_2 + s_1 = 25 \\
2. & \quad 3x_1 + x_2 + s_2 = 33 \\
3. & \quad 5x_1 + x_2 + s_3 = 45 \\
\end{align*}
\]
In this system of equations, \(x_1\) and \(x_2\) are the variables, while \(s_1\), \(s_2\), and \(s_3\) are the slack variables.
#### Basic Solutions:
A partially completed table is provided below showing ten solutions to the system of equations. Solutions (A) to (J) consist of various values assigned to \(x_1\), \(x_2\), \(s_1\), \(s_2\), and \(s_3\):
| Solutions | \(x_1\) | \(x_2\) | \(s_1\) | \(s_2\) | \(s_3\) |
|-----------|-----|-----|-----|-----|-----|
| (A) | 0 | 0 | 25 | 33 | 45 |
| (B) | 0 | 25 | 0 | 8 | 20 |
| (C) | 0 | 33 | -8 | 0 | 12 |
| (D) | 0 | 45 | -20 | -12 | 0 |
| (E) | 25 | 0 | 0 | -42 | -80 |
| (F) | 11 | 0 | 14 | -10 | 0 |
| (G) | 0 | 0 | 0 | 0 | 0 |
| (H) | 0 | 0 | 0 | 0 | 0 |
| (I) |
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