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The question asks how we can get System \( B \) from System \( A \). Let's analyze the two systems side by side:

### System \( A \):
\[
\begin{cases}
x - 4y = 1 \\
5x + 6y = -5
\end{cases}
\]

### System \( B \):
\[
\begin{cases}
x = 1 + 4y \\
5x + 6y = -5
\end{cases}
\]

In comparing the two systems:

1. **First equation in System \( A \):** \( x - 4y = 1 \)
2. **First equation in System \( B \):** \( x = 1 + 4y \)

Notice that the first equation in System \( B \) is simply a rearranged version of the first equation in System \( A \). Specifically, if we solve for \( x \) in System \( A \)'s first equation, we get:

\[
x - 4y = 1 \implies x = 1 + 4y
\]

Thus, the first equation in System \( B \) is obtained by rearranging the first equation of System \( A \) to isolate \( x \) on one side.

The second equation in both systems remains the same: \( 5x + 6y = -5 \).

Now, examining the multiple-choice options:

- **A: "Replace one equation with itself where a quantity is added to only one side"** – This is incorrect, as we did not add any new quantity; we only rearranged the existing terms.
- **B: "Replace one equation with itself where the same quantity is added to both sides"** – This is incorrect, as we did not add anything to both sides.
- **C: "Swap only the right-hand sides of both equations"** – This is incorrect because no swapping of the right-hand sides occurred.
- **D: "Swap the order of the equations"** – This is also incorrect since the order of the equations did not change.

### Correct answer:
The correct operation is rearranging the first equation to isolate \( x \). Therefore, the closest match is **A**, although the wording may be a bit ambiguous.

Would you like a deeper explanation or further clarification on this process?

Here are some related questions to extend your understanding:
1. How do you solve a system of linear equations by substitution?
2. What are the common methods for rearranging equations?
3. Can systems of equations have no solutions? Why or why not?
4. How do linear transformations affect systems of equations?
5. How can we graphically interpret the solutions of a system of equations?

**Tip:** When solving systems of equations, rearranging one equation can often simplify substitution into the other equation.

This problem demonstrates how to rearrange linear equations to isolate variables and compare systems. Learn how to transform System A into System B by rearranging the first equation, with full step-by-step solutions. Ideal for students in Grades 8-10.

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