https://file.aiquickdraw.com/mathbot/file/173303957289128o9gx4y.jpg

From the image, the problem involves selecting the correct row operation used to transform the matrix on the left into the matrix on the right. Let's analyze the situation step by step.

### Initial Matrix:
\[
\begin{bmatrix}
1 & 1 & -1 & | & 5 \\
5 & 0 & -20 & | & 15 \\
-2 & 2 & 3 & | & 10
\end{bmatrix}
\]

### Transformed Matrix:
\[
\begin{bmatrix}
1 & 1 & -1 & | & 5 \\
5 & 0 & -20 & | & 15 \\
0 & 4 & 1 & | & 20
\end{bmatrix}
\]

### Observing Changes:
The first two rows (\(R_1\) and \(R_2\)) remain unchanged. The operation modifies \(R_3\), which initially was:
\[
R_3 = [-2, 2, 3, | 10]
\]
and becomes:
\[
R_3 = [0, 4, 1, | 20]
\]

### Test Each Option:

1. **Option (a): \(-R_1 + 3R_3 \to R_3\)**  
   Performing this:
   \[
   -R_1 = [-1, -1, 1, | -5], \quad 3R_3 = [-6, 6, 9, | 30]
   \]
   Adding:
   \[
   -R_1 + 3R_3 = [-7, 5, 10, | 25] \quad \neq [0, 4, 1, | 20]
   \]
   Incorrect.

2. **Option (b): \(2R_1 + R_3 \to R_3\)**  
   Performing this:
   \[
   2R_1 = [2, 2, -2, | 10], \quad R_3 = [-2, 2, 3, | 10]
   \]
   Adding:
   \[
   2R_1 + R_3 = [0, 4, 1, | 20]
   \]
   Correct!

3. **Option (c): \(2R_3 - 3R_1 \to R_3\)**  
   Performing this:
   \[
   2R_3 = [-4, 4, 6, | 20], \quad -3R_1 = [-3, -3, 3, | -15]
   \]
   Adding:
   \[
   2R_3 - 3R_1 = [-7, 1, 9, | 5] \quad \neq [0, 4, 1, | 20]
   \]
   Incorrect.

4. **Option (d): \(-3R_2 + R_3 \to R_3\)**  
   Performing this:
   \[
   -3R_2 = [-15, 0, 60, | -45], \quad R_3 = [-2, 2, 3, | 10]
   \]
   Adding:
   \[
   -3R_2 + R_3 = [-17, 2, 63, | -35] \quad \neq [0, 4, 1, | 20]
   \]
   Incorrect.

### Final Answer:
**(b): \(2R_1 + R_3 \to R_3\)**

Would you like a detailed explanation of any step?  
Here are related questions for deeper understanding:

1. How do row operations affect the determinant of a matrix?  
2. What are common row reduction strategies for solving systems of equations?  
3. How can augmented matrices be used to find the inverse of a matrix?  
4. What are the properties of elementary row operations?  
5. How does Gaussian elimination relate to row operations?  

**Tip:** Always verify row operations by substituting values to ensure the transformation is correct.

Determine the correct row operation used to transform the given matrix.

Understand how to perform the correct row operation to transform a given matrix using Gaussian elimination. This detailed solution explores row transformations, including elementary operations like adding multiples of rows. Ideal for Linear Algebra learners.

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