Determine the value of h such that the matrix is the augmented matrix of a consistent linear system. 6 -8 | h] 24 4 -18 h =
Determine the value of h such that the matrix is the augmented matrix of a consistent linear system. 6 -8 | h] 24 4 -18 h =
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
![**Problem Statement:**
Determine the value of \( h \) such that the matrix is the augmented matrix of a consistent linear system.
\[
\begin{bmatrix}
6 & -8 & \vline & h \\
-18 & 24 & \vline & 4
\end{bmatrix}
\]
**Solution:**
To find the value of \( h \) that makes the system consistent, analyze the given matrix. The matrix represents a system of linear equations where consistency implies at least one solution exists. The consistency is related to the linear dependency or independency of the matrix’s rows.
**Calculation:**
1. The system of equations from the matrix can be expressed as:
\[
6x - 8y = h \\
-18x + 24y = 4
\]
2. To ensure consistency, check for a multiple relationship between the rows:
\[
\frac{-18}{6} = -3 \quad \text{and} \quad \frac{24}{-8} = -3
\]
This suggests the second row is a multiple of the first. Apply the same ratio to the constants to maintain consistency:
\[
\frac{4}{h} = -3
\]
3. Solve for \( h \):
\[
h = -\frac{4}{3}
\]
The consistent value of \( h \) is \(-\frac{4}{3}\).
**Conclusion:**
For the matrix to represent a consistent linear system, \( h \) must be \(-\frac{4}{3}\), ensuring the rows are proportionally consistent and do not lead to a contradiction.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70a05657-0307-4b7c-962c-ab3824846297%2F1ae98cc2-0580-4a80-88b6-ce481ea59154%2Fc4b8ff_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Determine the value of \( h \) such that the matrix is the augmented matrix of a consistent linear system.
\[
\begin{bmatrix}
6 & -8 & \vline & h \\
-18 & 24 & \vline & 4
\end{bmatrix}
\]
**Solution:**
To find the value of \( h \) that makes the system consistent, analyze the given matrix. The matrix represents a system of linear equations where consistency implies at least one solution exists. The consistency is related to the linear dependency or independency of the matrix’s rows.
**Calculation:**
1. The system of equations from the matrix can be expressed as:
\[
6x - 8y = h \\
-18x + 24y = 4
\]
2. To ensure consistency, check for a multiple relationship between the rows:
\[
\frac{-18}{6} = -3 \quad \text{and} \quad \frac{24}{-8} = -3
\]
This suggests the second row is a multiple of the first. Apply the same ratio to the constants to maintain consistency:
\[
\frac{4}{h} = -3
\]
3. Solve for \( h \):
\[
h = -\frac{4}{3}
\]
The consistent value of \( h \) is \(-\frac{4}{3}\).
**Conclusion:**
For the matrix to represent a consistent linear system, \( h \) must be \(-\frac{4}{3}\), ensuring the rows are proportionally consistent and do not lead to a contradiction.
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