https://file.aiquickdraw.com/mathbot/file/1729187124525x3wqgtw1.jpg

The augmented matrix Helena obtained is:

\[
\begin{bmatrix}
1 & 0 & 6 \\
0 & 0 & 0
\end{bmatrix}
\]

This matrix represents the system of linear equations after row operations. Let's interpret the rows:

1. The first row \( [1 \ 0 \ 6] \) corresponds to the equation \( x_1 = 6 \).
2. The second row \( [0 \ 0 \ 0] \) represents the equation \( 0 = 0 \), which is always true and does not provide additional information.

Given this, the system reduces to just \( x_1 = 6 \), and since there are no constraints on any other variables, the system has **infinitely many solutions** because at least one variable (e.g., \( x_2 \)) is free.

The correct answer is:
- **C. Infinitely many solutions**

Do you want more details or have any questions?

Here are 5 questions related to this explanation:
1. What would happen if the second row had non-zero elements?
2. How do free variables affect the number of solutions?
3. What is a consistent system of linear equations?
4. How does row reduction lead to different forms of solutions?
5. What is the difference between no solution and infinitely many solutions?

**Tip:** A row of zeros in an augmented matrix often indicates a free variable, leading to infinitely many solutions.

Helena used row operations on an augmented matrix representing a system of equations and obtained the following result: 

[1 0 6]
[0 0 0]

How many solutions does Helena’s system of equations have?
A. 0
B. 1
C. infinitely many
D. cannot be determined

Learn how to determine the number of solutions for a system of linear equations using an augmented matrix. In this example, the system has infinitely many solutions due to the presence of free variables in the row-reduced matrix.

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