Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![---
**Solve the problem.**
Find the general solution of the homogeneous system below. Give your answer as a vector.
\[
\begin{aligned}
x_1 + 2x_2 - 3x_3 &= 0 \\
4x_1 + 7x_2 - 9x_3 &= 0 \\
-x_1 - 4x_2 + 9x_3 &= 0
\end{aligned}
\]
\[
\begin{aligned}
\circ \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} =
\begin{bmatrix}
-3 \\
3 \\
1
\end{bmatrix}
\end{aligned}
\]
\[
\begin{aligned}
\circ \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} =
\begin{bmatrix}
-3 \\
3 \\
0
\end{bmatrix}
\end{aligned}
\]
\[
\begin{aligned}
\circ \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} =
\begin{bmatrix}
3 \\
-3 \\
1
\end{bmatrix}
\end{aligned}
\]
\[
\begin{aligned}
\circ \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} =
x_3 \begin{bmatrix}
-3 \\
3 \\
1
\end{bmatrix}
\end{aligned}
\]
---
This problem asks for the general solution of a given homogeneous system of linear equations and to express the result as a vector. There are four multiple-choice options provided, each offering a different possible solution vector. The correct answer should satisfy all three linear equations simultaneously.](https://content.bartleby.com/qna-images/question/157e66f5-e794-4af2-988e-1885b818a80a/824ce568-7b8f-47d2-abf6-c1affbcf0887/gpmlf5_processed.jpeg)
Transcribed Image Text:---
**Solve the problem.**
Find the general solution of the homogeneous system below. Give your answer as a vector.
\[
\begin{aligned}
x_1 + 2x_2 - 3x_3 &= 0 \\
4x_1 + 7x_2 - 9x_3 &= 0 \\
-x_1 - 4x_2 + 9x_3 &= 0
\end{aligned}
\]
\[
\begin{aligned}
\circ \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} =
\begin{bmatrix}
-3 \\
3 \\
1
\end{bmatrix}
\end{aligned}
\]
\[
\begin{aligned}
\circ \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} =
\begin{bmatrix}
-3 \\
3 \\
0
\end{bmatrix}
\end{aligned}
\]
\[
\begin{aligned}
\circ \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} =
\begin{bmatrix}
3 \\
-3 \\
1
\end{bmatrix}
\end{aligned}
\]
\[
\begin{aligned}
\circ \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} =
x_3 \begin{bmatrix}
-3 \\
3 \\
1
\end{bmatrix}
\end{aligned}
\]
---
This problem asks for the general solution of a given homogeneous system of linear equations and to express the result as a vector. There are four multiple-choice options provided, each offering a different possible solution vector. The correct answer should satisfy all three linear equations simultaneously.
![### Solving Systems of Linear Equations – Parametric Vector Form
#### Given Problem:
Describe all solutions of \( Ax = b \), where
\[ A = \begin{bmatrix}
2 & -5 & 3 \\
-2 & 6 & -5 \\
-4 & 7 & 0
\end{bmatrix} \]
and
\[ b = \begin{bmatrix}
-3 \\
4 \\
3
\end{bmatrix} \]
#### Task:
Describe the general solution in parametric vector form.
#### Solution Options:
1. Option 1:
\[ \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} =
\begin{bmatrix}
1 \\
1 \\
0
\end{bmatrix}
+ x_3
\begin{bmatrix}
7/2 \\
2 \\
0
\end{bmatrix} \]
2. Option 2:
\[ \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} =
\begin{bmatrix}
7/2 \\
2 \\
1
\end{bmatrix}
+ x_3
\begin{bmatrix}
1 \\
1 \\
0
\end{bmatrix} \]
3. Option 3:
\[ \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} =
\begin{bmatrix}
1 \\
1 \\
0
\end{bmatrix}
+ x_3
\begin{bmatrix}
7/2 \\
2 \\
1
\end{bmatrix} \]
4. Option 4:
\[ \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} =
\begin{bmatrix}
-3 \\
1 \\
0
\end{bmatrix}
+ x_3
\begin{bmatrix}
-1/2 \\
2 \\
1
\end{bmatrix} \]
Choose the correct option that represents the general solution in parametric vector form.](https://content.bartleby.com/qna-images/question/157e66f5-e794-4af2-988e-1885b818a80a/824ce568-7b8f-47d2-abf6-c1affbcf0887/3pl30sq_processed.jpeg)
Transcribed Image Text:### Solving Systems of Linear Equations – Parametric Vector Form
#### Given Problem:
Describe all solutions of \( Ax = b \), where
\[ A = \begin{bmatrix}
2 & -5 & 3 \\
-2 & 6 & -5 \\
-4 & 7 & 0
\end{bmatrix} \]
and
\[ b = \begin{bmatrix}
-3 \\
4 \\
3
\end{bmatrix} \]
#### Task:
Describe the general solution in parametric vector form.
#### Solution Options:
1. Option 1:
\[ \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} =
\begin{bmatrix}
1 \\
1 \\
0
\end{bmatrix}
+ x_3
\begin{bmatrix}
7/2 \\
2 \\
0
\end{bmatrix} \]
2. Option 2:
\[ \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} =
\begin{bmatrix}
7/2 \\
2 \\
1
\end{bmatrix}
+ x_3
\begin{bmatrix}
1 \\
1 \\
0
\end{bmatrix} \]
3. Option 3:
\[ \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} =
\begin{bmatrix}
1 \\
1 \\
0
\end{bmatrix}
+ x_3
\begin{bmatrix}
7/2 \\
2 \\
1
\end{bmatrix} \]
4. Option 4:
\[ \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} =
\begin{bmatrix}
-3 \\
1 \\
0
\end{bmatrix}
+ x_3
\begin{bmatrix}
-1/2 \\
2 \\
1
\end{bmatrix} \]
Choose the correct option that represents the general solution in parametric vector form.
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