**Linear Systems and Consistency Problems** **Exercises 15 and 16:** Determine if the systems are consistent. Do not completely solve the systems. 15. \[ \begin{cases} x_1 - 6x_2 = 5 \\ x_2 - 4x_3 + x_4 = 0 \\ -x_1 + 6x_2 + x_3 + 5x_4 = 3 \\ -x_2 + 5x_3 + 4x_4 = 0 \\ \end{cases} \] 16. \[ \begin{cases} 2x_1 - 4x_4 = -10 \\ 3x_2 + 3x_3 = 0 \\ x_3 + 4x_4 = -1 \\ -3x_1 + 2x_2 + 3x_3 + x_4 = 5 \\ \end{cases} \] **Exercise 17:** Do the three lines \(2x_1 + 3x_2 = -1\), \(6x_1 + 5x_2 = 0\), and \(2x_1 - 5x_2 = 7\) have a common point of intersection? Explain. **Exercise 18:** Do the three planes \(2x_1 + 4x_2 + 4x_3 = 4\), \(x_2 - 2x_3 = -2\), and \(2x_1 + 3x_2 = 0\) have at least one common point of intersection? Explain. **Exercises 19-22:** Determine the value(s) of \(h\) such that the matrix is the augmented matrix of a consistent linear system. 19. \[ \begin{bmatrix} 1 & h & 4 \\ 3 & 6 & 8 \\ \end{bmatrix} \] 20. \[ \begin{bmatrix} 1 & h & -5 \\ 2 & -8 & 6 \\ \end{bmatrix} \] 21. \[ \begin{bmatrix} 1 & 4 & -2 \\ 3 & h & -6 \\ \end{bmatrix} \] 22. \[ \begin{bmatrix} -4 & 12
**Linear Systems and Consistency Problems** **Exercises 15 and 16:** Determine if the systems are consistent. Do not completely solve the systems. 15. \[ \begin{cases} x_1 - 6x_2 = 5 \\ x_2 - 4x_3 + x_4 = 0 \\ -x_1 + 6x_2 + x_3 + 5x_4 = 3 \\ -x_2 + 5x_3 + 4x_4 = 0 \\ \end{cases} \] 16. \[ \begin{cases} 2x_1 - 4x_4 = -10 \\ 3x_2 + 3x_3 = 0 \\ x_3 + 4x_4 = -1 \\ -3x_1 + 2x_2 + 3x_3 + x_4 = 5 \\ \end{cases} \] **Exercise 17:** Do the three lines \(2x_1 + 3x_2 = -1\), \(6x_1 + 5x_2 = 0\), and \(2x_1 - 5x_2 = 7\) have a common point of intersection? Explain. **Exercise 18:** Do the three planes \(2x_1 + 4x_2 + 4x_3 = 4\), \(x_2 - 2x_3 = -2\), and \(2x_1 + 3x_2 = 0\) have at least one common point of intersection? Explain. **Exercises 19-22:** Determine the value(s) of \(h\) such that the matrix is the augmented matrix of a consistent linear system. 19. \[ \begin{bmatrix} 1 & h & 4 \\ 3 & 6 & 8 \\ \end{bmatrix} \] 20. \[ \begin{bmatrix} 1 & h & -5 \\ 2 & -8 & 6 \\ \end{bmatrix} \] 21. \[ \begin{bmatrix} 1 & 4 & -2 \\ 3 & h & -6 \\ \end{bmatrix} \] 22. \[ \begin{bmatrix} -4 & 12
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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Topic Video
Question
Number 16 1.1
![**Linear Systems and Consistency Problems**
**Exercises 15 and 16:** Determine if the systems are consistent. Do not completely solve the systems.
15.
\[
\begin{cases}
x_1 - 6x_2 = 5 \\
x_2 - 4x_3 + x_4 = 0 \\
-x_1 + 6x_2 + x_3 + 5x_4 = 3 \\
-x_2 + 5x_3 + 4x_4 = 0 \\
\end{cases}
\]
16.
\[
\begin{cases}
2x_1 - 4x_4 = -10 \\
3x_2 + 3x_3 = 0 \\
x_3 + 4x_4 = -1 \\
-3x_1 + 2x_2 + 3x_3 + x_4 = 5 \\
\end{cases}
\]
**Exercise 17:** Do the three lines \(2x_1 + 3x_2 = -1\), \(6x_1 + 5x_2 = 0\), and \(2x_1 - 5x_2 = 7\) have a common point of intersection? Explain.
**Exercise 18:** Do the three planes \(2x_1 + 4x_2 + 4x_3 = 4\), \(x_2 - 2x_3 = -2\), and \(2x_1 + 3x_2 = 0\) have at least one common point of intersection? Explain.
**Exercises 19-22:** Determine the value(s) of \(h\) such that the matrix is the augmented matrix of a consistent linear system.
19.
\[
\begin{bmatrix}
1 & h & 4 \\
3 & 6 & 8 \\
\end{bmatrix}
\]
20.
\[
\begin{bmatrix}
1 & h & -5 \\
2 & -8 & 6 \\
\end{bmatrix}
\]
21.
\[
\begin{bmatrix}
1 & 4 & -2 \\
3 & h & -6 \\
\end{bmatrix}
\]
22.
\[
\begin{bmatrix}
-4 & 12](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee22cf2f-b974-4b00-a3cf-09b388e7d65d%2Fc996037d-3fa9-4b1e-8286-0f462c0f00fb%2F0uxtbrj.jpeg&w=3840&q=75)
Transcribed Image Text:**Linear Systems and Consistency Problems**
**Exercises 15 and 16:** Determine if the systems are consistent. Do not completely solve the systems.
15.
\[
\begin{cases}
x_1 - 6x_2 = 5 \\
x_2 - 4x_3 + x_4 = 0 \\
-x_1 + 6x_2 + x_3 + 5x_4 = 3 \\
-x_2 + 5x_3 + 4x_4 = 0 \\
\end{cases}
\]
16.
\[
\begin{cases}
2x_1 - 4x_4 = -10 \\
3x_2 + 3x_3 = 0 \\
x_3 + 4x_4 = -1 \\
-3x_1 + 2x_2 + 3x_3 + x_4 = 5 \\
\end{cases}
\]
**Exercise 17:** Do the three lines \(2x_1 + 3x_2 = -1\), \(6x_1 + 5x_2 = 0\), and \(2x_1 - 5x_2 = 7\) have a common point of intersection? Explain.
**Exercise 18:** Do the three planes \(2x_1 + 4x_2 + 4x_3 = 4\), \(x_2 - 2x_3 = -2\), and \(2x_1 + 3x_2 = 0\) have at least one common point of intersection? Explain.
**Exercises 19-22:** Determine the value(s) of \(h\) such that the matrix is the augmented matrix of a consistent linear system.
19.
\[
\begin{bmatrix}
1 & h & 4 \\
3 & 6 & 8 \\
\end{bmatrix}
\]
20.
\[
\begin{bmatrix}
1 & h & -5 \\
2 & -8 & 6 \\
\end{bmatrix}
\]
21.
\[
\begin{bmatrix}
1 & 4 & -2 \\
3 & h & -6 \\
\end{bmatrix}
\]
22.
\[
\begin{bmatrix}
-4 & 12
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