To determine whether each pair of lines is parallel, perpendicular, or neither, we need to analyze the slopes of the lines. I’ll solve step by step.

Line 1: $3y = -4x + 2$

1. Rearrange to slope-intercept form $y = mx + b$:
   $y = -\frac{4}{3}x + \frac{2}{3}$
2. Slope ($m_1$) of Line 1:
   $m_1 = -\frac{4}{3}$

Line 2: $y = -\frac{4}{3}x + 7$

1. Already in slope-intercept form $y = mx + b$.
2. Slope ($m_2$) of Line 2:
   $m_2 = -\frac{4}{3}$

Line 3: $6x - 8y = 8$

1. Rearrange to slope-intercept form $y = mx + b$:
   $-8y = -6x + 8 \quad \Rightarrow \quad y = \frac{3}{4}x - 1$
2. Slope ($m_3$) of Line 3:
   $m_3 = \frac{3}{4}$

Compare the Slopes:

1. Line 1 and Line 2:
   Slopes $m_1 = -\frac{4}{3}$ and $m_2 = -\frac{4}{3}$.

   • Slopes are equal, so the lines are parallel.

2. Line 1 and Line 3:
   Slopes $m_1 = -\frac{4}{3}$ and $m_3 = \frac{3}{4}$.

   • Check if they are perpendicular: Multiply the slopes:
     $m_1 \cdot m_3 = -\frac{4}{3} \cdot \frac{3}{4} = -1$
     • Product equals $-1$, so the lines are perpendicular.

3. Line 2 and Line 3:
   Slopes $m_2 = -\frac{4}{3}$ and $m_3 = \frac{3}{4}$.

   • Check if they are perpendicular:
     $m_2 \cdot m_3 = -\frac{4}{3} \cdot \frac{3}{4} = -1$
     • Product equals $-1$, so the lines are perpendicular.

Final Answers:

• Line 1 and Line 2: Parallel
• Line 1 and Line 3: Perpendicular
• Line 2 and Line 3: Perpendicular

Let me know if you need further explanation or want me to clarify any steps!

Related Questions:

1. What does it mean when two lines are perpendicular geometrically?
2. How do you determine the slope of a line from its equation in standard form?
3. Can two lines with the same slopes have different intercepts?
4. What happens if two lines have slopes whose product is not $-1$?
5. How do parallel and perpendicular lines relate to triangles in geometry?

Tip: The slope-intercept form $y = mx + b$ is the quickest way to identify the slope of a line.