Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![## Identifying Cases with No Solutions in a Linear System
In the given problem, you need to determine the value of \( k \) for which the following system of linear equations has no solutions:
\[ \begin{bmatrix}
1 & 1 & 3 & | & -2 \\
1 & 2 & -5 & | & 2 \\
6 & 14 & k & | & 21
\end{bmatrix} \]
This represents a system of three linear equations with three variables. To solve it, let's set up the augmented matrix and perform row operations to find the value of \( k \) that makes the system inconsistent (i.e., when it has no solutions).
**Explore the Augmented Matrix:**
\[ \begin{bmatrix}
1 & 1 & 3 & -2 \\
1 & 2 & -5 & 2 \\
6 & 14 & k & 21
\end{bmatrix} \]
**Determine the Condition for No Solutions:**
1. Start with elementary row operations to simplify the matrix.
2. Identify any inconsistencies, such as a row that translates to a false statement (e.g., \( 0 = 1 \)).
**For what value of \( k \) will the system have no solutions?**
\[ k = \boxed{\text{ }}\]
This question focuses on solving the augmented matrix to identify the condition under which the system of equations becomes inconsistent, leading to no solutions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e366a85-c23b-450a-9bc7-d2ea0be52d3f%2Fcf0aa920-a2e3-49ce-871d-37c4eb051241%2Fkyrtv7c_processed.png&w=3840&q=75)
Transcribed Image Text:## Identifying Cases with No Solutions in a Linear System
In the given problem, you need to determine the value of \( k \) for which the following system of linear equations has no solutions:
\[ \begin{bmatrix}
1 & 1 & 3 & | & -2 \\
1 & 2 & -5 & | & 2 \\
6 & 14 & k & | & 21
\end{bmatrix} \]
This represents a system of three linear equations with three variables. To solve it, let's set up the augmented matrix and perform row operations to find the value of \( k \) that makes the system inconsistent (i.e., when it has no solutions).
**Explore the Augmented Matrix:**
\[ \begin{bmatrix}
1 & 1 & 3 & -2 \\
1 & 2 & -5 & 2 \\
6 & 14 & k & 21
\end{bmatrix} \]
**Determine the Condition for No Solutions:**
1. Start with elementary row operations to simplify the matrix.
2. Identify any inconsistencies, such as a row that translates to a false statement (e.g., \( 0 = 1 \)).
**For what value of \( k \) will the system have no solutions?**
\[ k = \boxed{\text{ }}\]
This question focuses on solving the augmented matrix to identify the condition under which the system of equations becomes inconsistent, leading to no solutions.
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