### Systems of Equations and Basic SolutionsThe 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) |

The partially completed table below shows the ten basic solutions to the following e-system. The basic variable(s) in basic solution (D) is/are (Use a comma to separate answers as needed.) X, + X2 + S, = 25 3x, + X2 - 33 5x, + X2 *S3 = 45 In basic solution (D), which variables are basic? X2 S2 (A) (В) (C) (D) (E) (F) (G) (H) (1) (J) 25 33 45 25 20 33 - 8 12 - 12 - 42 45 -20 25 - 80 11 14 - 10

The partially completed table below shows the ten basic solutions to the following e-system. The basic variable(s) in basic solution (D) is/are (Use a comma to separate answers as needed.) X, + X2 + S, = 25 3x, + X2 - 33 5x, + X2 *S3 = 45 In basic solution (D), which variables are basic? X2 S2 (A) (В) (C) (D) (E) (F) (G) (H) (1) (J) 25 33 45 25 20 33 - 8 12 - 12 - 42 45 -20 25 - 80 11 14 - 10

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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Question

### 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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