https://file.aiquickdraw.com/mathbot/file/1729444012424x8s96vdw.jpg

The given problem involves determining the solution set for a system of linear equations based on the augmented matrix:

\[
\begin{pmatrix}
1 & 3 & | & 5 \\
2 & 6 & | & 11
\end{pmatrix}
\]

### Step 1: Write the system of linear equations
The augmented matrix represents the following system of equations:
\[
1x + 3y = 5 \quad \text{(Equation 1)}
\]
\[
2x + 6y = 11 \quad \text{(Equation 2)}
\]

### Step 2: Solve the system using elimination
We can use the elimination method by multiplying Equation 1 by 2 and subtracting it from Equation 2 to eliminate \(x\):
\[
2(x + 3y) = 2(5) \quad \Rightarrow \quad 2x + 6y = 10
\]
Now, subtract this from Equation 2:
\[
(2x + 6y) - (2x + 6y) = 11 - 10
\]
\[
0 = 1
\]

This is a contradiction, meaning the system is inconsistent and has no solution.

### Conclusion:
The correct answer is **d) has no solution**.

Would you like a more detailed explanation or have any questions?

Here are 5 related questions for further exploration:
1. How do you recognize an inconsistent system from its augmented matrix?
2. What is the geometric interpretation of a system that has no solution?
3. How would the solution change if the augmented matrix were different?
4. Can a system with more equations than variables still have no solution?
5. How is row-reduction related to solving systems of equations?

**Tip:** Always check for inconsistencies when using elimination, especially if you encounter equations like \(0 = 1\), which signal no solution.

The linear system corresponding to the augmented matrix: 
[1 3 | 5 
 2 6 | 11] 
 a) has infinitely many solutions 
 b) has a unique solution 
 c) has two solutions 
 d) has no solution 
 e) None of the above

Explore how to solve a system of linear equations using an augmented matrix and Gaussian elimination. Learn why the given system has no solution, with detailed steps in identifying inconsistencies. Suitable for Grades 10-12.

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