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To solve the problem, we need to identify lines that are either **parallel** or **perpendicular** to the given line \( A \), which has the equation:

\[
y = 4x - 3
\]

### 1. **Parallel Lines:**
Lines are **parallel** if they have the same slope. The slope-intercept form of a line is \( y = mx + b \), where \( m \) is the slope. For line \( A \), the slope is \( m = 4 \).

Thus, any line with slope \( 4 \) is parallel to line \( A \).

### 2. **Perpendicular Lines:**
Lines are **perpendicular** if the product of their slopes is \( -1 \). So, for a line to be perpendicular to line \( A \), its slope \( m' \) must satisfy:

\[
4 \times m' = -1 \quad \Rightarrow \quad m' = -\frac{1}{4}
\]

Thus, any line with slope \( -\frac{1}{4} \) is perpendicular to line \( A \).

### Now, let's analyze the given equations:

1. \( y = 4x + 2 \) ⟶ Slope is \( 4 \) (Parallel to line \( A \)).
2. \( y = -\frac{1}{4}x + 5 \) ⟶ Slope is \( -\frac{1}{4} \) (Perpendicular to line \( A \)).
3. \( y = 4x + 9 \) ⟶ Slope is \( 4 \) (Parallel to line \( A \)).
4. \( y + 4 = -\frac{1}{4}(x - 3) \) ⟶ This can be rewritten as \( y = -\frac{1}{4}x + \frac{1}{4} \), so slope is \( -\frac{1}{4} \) (Perpendicular to line \( A \)).
5. \( y = -4x + 6 \) ⟶ Slope is \( -4 \) (Neither parallel nor perpendicular).
6. \( y - 4 = 4(x + 3) \) ⟶ This can be rewritten as \( y = 4x + 16 \), so slope is \( 4 \) (Parallel to line \( A \)).
7. \( y = \frac{1}{4}x + 8 \) ⟶ Slope is \( \frac{1}{4} \) (Neither parallel nor perpendicular).
8. \( y = -\frac{1}{4}x + 9 \) ⟶ Slope is \( -\frac{1}{4} \) (Perpendicular to line \( A \)).

### Final Sorting:

- **Parallel to Line \( A \):**
  - \( y = 4x + 2 \)
  - \( y = 4x + 9 \)
  - \( y - 4 = 4(x + 3) \) ⟶ \( y = 4x + 16 \)

- **Perpendicular to Line \( A \):**
  - \( y = -\frac{1}{4}x + 5 \)
  - \( y + 4 = -\frac{1}{4}(x - 3) \) ⟶ \( y = -\frac{1}{4}x + \frac{1}{4} \)
  - \( y = -\frac{1}{4}x + 9 \)

Would you like any further details or explanations?

Here are some related questions:
1. What are the general conditions for two lines to be perpendicular in 3D space?
2. How does the slope-intercept form of a line differ from the point-slope form?
3. Can lines with different slopes ever be parallel?
4. What are the conditions for lines to be coincident?
5. How does the slope of a line change when reflecting it across the x-axis?

**Tip:** Always remember, parallel lines share the same slope, and perpendicular lines' slopes multiply to give \( -1 \).

Sort the following equations of lines as parallel or perpendicular to line A, where line A has the equation y = 4x - 3.

Learn how to sort equations of lines that are parallel or perpendicular to a given line. This problem involves analyzing the slope-intercept form of equations and applying the concepts of parallel and perpendicular lines. Suitable for students in Grades 8-10.

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