Determine the value(s) of k such that the system of linear equations has the indicated number of solutions. (Enter your answers as a comma-separated list.) Infinitely many solutions 16x + ky = 20 kx + y = -5 k = 4, – 4
Determine the value(s) of k such that the system of linear equations has the indicated number of solutions. (Enter your answers as a comma-separated list.) Infinitely many solutions 16x + ky = 20 kx + y = -5 k = 4, – 4
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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![### Determine the value(s) of \( k \) such that the system of linear equations has the indicated number of solutions. (Enter your answers as a comma-separated list.)
**Infinitely many solutions**
\[
\begin{aligned}
16x + ky &= 20 \\
kx + y &= -5
\end{aligned}
\]
\[ k = [ \text{input box: 4, -4 } ] \quad \textcolor{red}{\text{X}} \]
### Explanation:
The problem asks for the value(s) of \( k \) that result in the system of linear equations having infinitely many solutions. The given system of equations is:
1. \( 16x + ky = 20 \)
2. \( kx + y = -5 \)
To have infinitely many solutions, the system of linear equations needs to be dependent, meaning one equation must be a scalar multiple of the other.
From the calculations, it is found that the correct values of \( k \) that lead to infinitely many solutions are \(-4\) and \(-\frac{125}{16}\). Therefore, the value(s) entered in the input box are incorrect.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F73fb2883-df73-4e37-b7ef-66d24e1d1095%2Fcda16aba-ba17-46de-9a3d-4e33da7f144c%2Fohwbrj_processed.png&w=3840&q=75)
Transcribed Image Text:### Determine the value(s) of \( k \) such that the system of linear equations has the indicated number of solutions. (Enter your answers as a comma-separated list.)
**Infinitely many solutions**
\[
\begin{aligned}
16x + ky &= 20 \\
kx + y &= -5
\end{aligned}
\]
\[ k = [ \text{input box: 4, -4 } ] \quad \textcolor{red}{\text{X}} \]
### Explanation:
The problem asks for the value(s) of \( k \) that result in the system of linear equations having infinitely many solutions. The given system of equations is:
1. \( 16x + ky = 20 \)
2. \( kx + y = -5 \)
To have infinitely many solutions, the system of linear equations needs to be dependent, meaning one equation must be a scalar multiple of the other.
From the calculations, it is found that the correct values of \( k \) that lead to infinitely many solutions are \(-4\) and \(-\frac{125}{16}\). Therefore, the value(s) entered in the input box are incorrect.
![### Solving a System of Equations by Substitution
**Objective:**
Solve the system of equations by letting \( A = \frac{1}{x} \) and \( B = \frac{1}{y} \). If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, set \( y = t \) and solve for \( x \) in terms of \( t \).
**System of Equations:**
\[
\frac{12}{x} - \frac{12}{y} = 7
\]
\[
\frac{4}{x} + \frac{3}{y} = 0
\]
**Solution Process:**
1. **Transform Variables:**
- Let \( A = \frac{1}{x} \)
- Let \( B = \frac{1}{y} \)
2. **Substitute:**
- Rewrite the system using \( A \) and \( B \):
\[
12A - 12B = 7
\]
\[
4A + 3B = 0
\]
3. **Solve the Transformed System:**
- The solution involves solving the linear equations for \( A \) and \( B \):
4. **Back-Substitute:**
- Once \( A \) and \( B \) are found, convert them back to \( x \) and \( y \) using \( x = \frac{1}{A} \) and \( y = \frac{1}{B} \).
**Example Solution:**
If we go through the process and derive the following solution:
\[
(x, y) = \left( \frac{21}{7}, -4 \right)
\]
**Final Answer:**
However, in this case, the provided solution box with the value \( \left( \frac{21}{7}, -4 \right) \) is marked with a red 'X', indicating that the solution is incorrect.
**Conclusion:**
Upon reviewing the problem solving steps and the substitution process, double-checking the steps or re-sloving using another method may be required to find the correct solution. If the system has no solution despite multiple verification attempts, it should be marked as "NO SOLUTION."](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F73fb2883-df73-4e37-b7ef-66d24e1d1095%2Fcda16aba-ba17-46de-9a3d-4e33da7f144c%2F48yzcrq_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving a System of Equations by Substitution
**Objective:**
Solve the system of equations by letting \( A = \frac{1}{x} \) and \( B = \frac{1}{y} \). If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, set \( y = t \) and solve for \( x \) in terms of \( t \).
**System of Equations:**
\[
\frac{12}{x} - \frac{12}{y} = 7
\]
\[
\frac{4}{x} + \frac{3}{y} = 0
\]
**Solution Process:**
1. **Transform Variables:**
- Let \( A = \frac{1}{x} \)
- Let \( B = \frac{1}{y} \)
2. **Substitute:**
- Rewrite the system using \( A \) and \( B \):
\[
12A - 12B = 7
\]
\[
4A + 3B = 0
\]
3. **Solve the Transformed System:**
- The solution involves solving the linear equations for \( A \) and \( B \):
4. **Back-Substitute:**
- Once \( A \) and \( B \) are found, convert them back to \( x \) and \( y \) using \( x = \frac{1}{A} \) and \( y = \frac{1}{B} \).
**Example Solution:**
If we go through the process and derive the following solution:
\[
(x, y) = \left( \frac{21}{7}, -4 \right)
\]
**Final Answer:**
However, in this case, the provided solution box with the value \( \left( \frac{21}{7}, -4 \right) \) is marked with a red 'X', indicating that the solution is incorrect.
**Conclusion:**
Upon reviewing the problem solving steps and the substitution process, double-checking the steps or re-sloving using another method may be required to find the correct solution. If the system has no solution despite multiple verification attempts, it should be marked as "NO SOLUTION."
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