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Let's solve the system of equations using augmented matrix methods. The system provided is:

\[
2x_1 - x_2 = 7
\]
\[
x_1 + 5x_2 = 20
\]

### Step 1: Set up the augmented matrix
The system of equations can be represented as an augmented matrix:

\[
\begin{pmatrix}
2 & -1 & | & 7 \\
1 & 5 & | & 20
\end{pmatrix}
\]

### Step 2: Perform row operations to get the matrix into row-echelon form

1. Leave the first row as it is:
   \[
   R_1: \quad 2 \quad -1 \quad | \quad 7
   \]

2. Use row operations to eliminate \( x_1 \) from the second row. Multiply \( R_2 \) by 2 and subtract \( R_1 \):

   \[
   R_2 \rightarrow 2R_2 - R_1 = (2 \cdot 1, 2 \cdot 5, 2 \cdot 20) - (2, -1, 7)
   \]

   This gives:
   \[
   0 \quad 11 \quad | \quad 33
   \]

Now the augmented matrix is:
\[
\begin{pmatrix}
2 & -1 & | & 7 \\
0 & 11 & | & 33
\end{pmatrix}
\]

### Step 3: Solve for the variables

From the second row, we can solve for \( x_2 \):
\[
11x_2 = 33 \quad \Rightarrow \quad x_2 = \frac{33}{11} = 3
\]

Substitute \( x_2 = 3 \) into the first row equation:
\[
2x_1 - 3 = 7 \quad \Rightarrow \quad 2x_1 = 10 \quad \Rightarrow \quad x_1 = \frac{10}{2} = 5
\]

### Final solution:
The unique solution is:
\[
x_1 = 5, \quad x_2 = 3
\]

Thus, the correct answer is **A**: The unique solution is \( x_1 = 5 \) and \( x_2 = 3 \).

---

Do you want further details or have any questions? 

Here are 5 related questions:
1. How do we know when a system has no solutions using augmented matrices?
2. How can you tell if a system has infinitely many solutions using row reduction?
3. What are the steps to convert a system to row-reduced echelon form?
4. How can augmented matrices be used for larger systems (3x3 or more)?
5. What do inconsistent rows indicate in terms of the system of equations?

**Tip**: Always check the determinant of the coefficient matrix to see if the system is consistent.

Solve the system of equations using augmented matrix methods: 2x1 - x2 = 7 and x1 + 5x2 = 20.

Learn how to solve the system of linear equations 2x1 - x2 = 7 and x1 + 5x2 = 20 using augmented matrix methods. This problem involves setting up the augmented matrix, performing row operations, and solving for the variables using back substitution. Suitable for high school or college-level students.

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