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Gladys is trying to convert the given matrix into row echelon form. To do so, she needs to apply appropriate elementary row operations.

The matrix provided is:
\[
\begin{pmatrix}
2 & -3 & -1 & -19 \\
-1 & -2 & 1 & 14 \\
1 & 3 & 5 & 19
\end{pmatrix}
\]

Row echelon form requires that the matrix have leading ones (pivots) and zeros below the pivots. Let's analyze the elementary row operations in the multiple-choice answers:

- **A. Add row #3 to row #1 to zero out the -19 there.**
  - This operation would modify the last column, which isn't the most immediate step needed. We generally aim to get zeros in the first column first.

- **B. Add row #3 to row #2 to zero out the -1 there.**
  - Adding row #3 to row #2 would help reduce the first column by removing the -1 in the second row, helping us toward row echelon form. This operation is useful.

- **C. Add row #2 to row #3 to zero out the 1 there.**
  - This operation is less useful early in the process. The goal should be to first clear out numbers below the leading 2 in the first column.

- **D. Multiply row #2 by 2 to make the first entries in rows #1 and #2 additive inverses.**
  - This is an excellent choice. By multiplying row #2 by 2, you can then add it to row #1, zeroing out the first column in the second row.

- **E. Multiply row #1 by 5 to make the third entries in rows #1 and #3 additive inverses.**
  - While this could be useful later for clearing the third column, it isn't the most direct step toward row echelon form right now.

### The best operation to help achieve row echelon form immediately would be **D**.

Would you like more details on any of the steps?

Here are 5 relative questions for further exploration:
1. What are the main characteristics of a matrix in row echelon form?
2. How do we handle matrices when leading entries are zero?
3. What is the difference between row echelon form and reduced row echelon form?
4. Can elementary row operations affect the solution to a system of equations?
5. How would the solution change if the matrix had more rows or columns?

**Tip:** Always prioritize operations that simplify the leftmost pivot first, then move systematically to the right.

Gladys has rewritten a system of equations as a matrix and wants to convert it into row echelon form. Which elementary row operations will help her achieve row echelon form?

Learn how to convert a system of equations represented as a matrix into row echelon form using elementary row operations. This problem includes key concepts from linear algebra, such as row operations and matrix transformations, and is ideal for students in Grades 10-12.

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