To determine which lines are parallel or perpendicular to line A, we need to compare their slopes.

Line A equation: $y = 4x - 3$

• The slope of line A is 4.

For parallel lines:

• Lines with the same slope as line A (i.e., slope = 4) are parallel.

For perpendicular lines:

• Lines whose slopes are the negative reciprocal of the slope of line A (i.e., slope = -1/4) are perpendicular.

Now, let's identify the slope of each equation:

1. $y = 4x + 2$

   • Slope = 4
     Parallel to line A

2. $y = -14x + 5$

   • Slope = -14
     Neither parallel nor perpendicular

3. $y = 4x + 9$

   • Slope = 4
     Parallel to line A

4. $y + 4 = -14(x - 3)$

   • Rewrite as: $y = -14x + 38$
   • Slope = -14
     Neither parallel nor perpendicular

5. $y = -4x + 6$

   • Slope = -4
     Neither parallel nor perpendicular

6. $y - 4 = 4(x + 3)$

   • Rewrite as: $y = 4x + 16$
   • Slope = 4
     Parallel to line A

7. $y = 14x + 8$

   • Slope = 14
     Neither parallel nor perpendicular

8. $y = -14x + 9$

   • Slope = -14
     Neither parallel nor perpendicular

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Summary:

Parallel to Line A:

• $y = 4x + 2$
• $y = 4x + 9$
• $y - 4 = 4(x + 3)$

Perpendicular to Line A:

• None of the equations have a slope of -1/4, so no perpendicular lines.

Would you like more details or have any questions?

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Here are five related questions to expand on this topic:

1. How do you determine the slope of a line from an equation?
2. Can two lines with different y-intercepts be parallel?
3. What is the geometric relationship between parallel and perpendicular lines?
4. How can we transform a given equation into slope-intercept form $y = mx + b$?
5. What are other methods to check if two lines are perpendicular?

Tip: Always rewrite equations into slope-intercept form $y = mx + b$ to easily identify the slope.