Esther rewrites a system of equations of variables x, y, and z as a matrix, which she then starts to reduce. She stops when she has the result below: 193−203150006This matrix is not in reduced row echelon form yet, so why did Esther stop?A.Esther realized that this system must have no solutions.B.Esther realized she can rewrite the system and solve it by substitution.C.Esther realized that this is no longer an augmented matrix so she must have made a mistake.D.Esther realized that this system must have infinitely many solutions.

To solve this problem, we should analyze the matrix:

\[
\begin{bmatrix}
1 & 9 & 3 & -20 \\
3 & 15 & 0 & 0 \\
0 & 0 & 6 & 6 \\
\end{bmatrix}
\]

This matrix represents a system of linear equations. Each row corresponds to an equation, and the columns represent coefficients for the variables \(x\), \(y\), and \(z\), with the last column being the constants from the equations.

### Step-by-Step Analysis:

1. **First Row: \(1x + 9y + 3z = -20\)**  
   The first row can be interpreted as the equation \(x + 9y + 3z = -20\).

2. **Second Row: \(3x + 15y = 0\)**  
   The second row represents the equation \(3x + 15y = 0\), indicating a relationship between \(x\) and \(y\).

3. **Third Row: \(6z = 6\)**  
   The third row simplifies to \(z = 1\).

### Why Did Esther Stop?

- The third row already gives a direct solution for \(z\), meaning \(z = 1\).
- The second row can be simplified to \(x + 5y = 0\) (by dividing through by 3), allowing substitution to find \(x\) in terms of \(y\).
- Since there are free variables (such as \(y\)) that can take any value, this indicates that the system has infinitely many solutions.

Thus, Esther stopped because she realized that the system must have infinitely many solutions.

### Correct Answer:
**D. Esther realized that this system must have infinitely many solutions.**

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

### Related Questions:
1. What is the difference between row echelon form and reduced row echelon form?
2. How can you tell if a system of linear equations has no solution, one solution, or infinitely many solutions?
3. What does it mean when a row in a matrix has all zeros except for the last column?
4. How do you perform row reduction to achieve reduced row echelon form?
5. What role do free variables play in determining the number of solutions for a system?

### Tip:
When reducing a matrix, always check for rows that simplify to a contradiction (like \(0 = 1\)) to identify if the system has no solutions.

Esther rewrites a system of equations of variables x, y, and z as a matrix, which she then starts to reduce. She stops when she has the result below: 1 9 3 -20 3 15 0 0 0 0 6 6. This matrix is not in reduced row echelon form yet, so why did Esther stop? A. Esther realized that this system must have no solutions. B. Esther realized she can rewrite the system and solve it by substitution. C. Esther realized that this is no longer an augmented matrix so she must have made a mistake. D. Esther realized that this system must have infinitely many solutions.

Learn how to analyze a system of equations using matrix row reduction techniques. This problem covers key concepts in linear algebra, including row echelon form and the conditions for systems with infinitely many solutions. Suitable for Grades 10-12.

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