In Exercises 9-12, solve the lin the b's solve the systems toget Ma ond
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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F1.6 Question 9 on paper
![### Linear Systems and Row Echelon Form
In this section, we'll explore solving linear systems by reducing an augmented matrix to its reduced row echelon form. You will practice these skills in the exercises provided.
#### Example
Consider the linear system represented in matrix form:
\[
\begin{pmatrix}
1 & 2 & 3 \\
1 & 5 & -5 \\
3 & 5 & 8
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}
=
\begin{pmatrix}
b_1 \\
b_2 \\
b_3
\end{pmatrix}
\]
**Exercises 7 and 8: Solve the linear systems given.**
7.
\[
\begin{cases}
3x_1 + 5x_2 = b_1 \\
x_1 + 2x_2 = b_2
\end{cases}
\]
8.
\[
\begin{cases}
x_1 + 2x_2 + 3x_3 = b_1 \\
2x_1 + 5x_2 - 5x_3 = b_2 \\
3x_1 + 5x_2 + 8x_3 = b_3
\end{cases}
\]
**In Exercises 9-12, solve the linear systems. Using the given values for the \(b\)'s solve the systems together by reducing an appropriate augmented matrix to reduced row echelon form.**
9.
\[
\begin{cases}
x_1 - 5x_2 = b_1 \\
3x_1 + 2x_2 = b_2
\end{cases}
\]
For part (i) \(b_1 = 1, b_2 = 4\)
For part (ii) \(b_1 = -2, b_2 = 5\)
10.
\[
\begin{cases}
-x_1 + 4x_2 + x_3 = b_1 \\
x_1 + 9x_2 - 2x_3 = b_2 \\
6x_1 + 4x_2 - 8x_3 = b_3
\end{cases}
\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fccdd90b5-cd41-4340-8a36-15b102c9ba73%2Fb7b1aba8-a693-4955-8591-332b7dd7ebdc%2Fn1l1v9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Systems and Row Echelon Form
In this section, we'll explore solving linear systems by reducing an augmented matrix to its reduced row echelon form. You will practice these skills in the exercises provided.
#### Example
Consider the linear system represented in matrix form:
\[
\begin{pmatrix}
1 & 2 & 3 \\
1 & 5 & -5 \\
3 & 5 & 8
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}
=
\begin{pmatrix}
b_1 \\
b_2 \\
b_3
\end{pmatrix}
\]
**Exercises 7 and 8: Solve the linear systems given.**
7.
\[
\begin{cases}
3x_1 + 5x_2 = b_1 \\
x_1 + 2x_2 = b_2
\end{cases}
\]
8.
\[
\begin{cases}
x_1 + 2x_2 + 3x_3 = b_1 \\
2x_1 + 5x_2 - 5x_3 = b_2 \\
3x_1 + 5x_2 + 8x_3 = b_3
\end{cases}
\]
**In Exercises 9-12, solve the linear systems. Using the given values for the \(b\)'s solve the systems together by reducing an appropriate augmented matrix to reduced row echelon form.**
9.
\[
\begin{cases}
x_1 - 5x_2 = b_1 \\
3x_1 + 2x_2 = b_2
\end{cases}
\]
For part (i) \(b_1 = 1, b_2 = 4\)
For part (ii) \(b_1 = -2, b_2 = 5\)
10.
\[
\begin{cases}
-x_1 + 4x_2 + x_3 = b_1 \\
x_1 + 9x_2 - 2x_3 = b_2 \\
6x_1 + 4x_2 - 8x_3 = b_3
\end{cases}
\
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