To determine whether each pair of lines is parallel, perpendicular, or neither, we need to find the slopes of the lines. Two lines are parallel if their slopes are equal, and perpendicular if the product of their slopes is $-1$.

Let's analyze each line:

1. Find the slope of each line:

Line 1: $10x - 6y = -4$

To find the slope, we'll rewrite this in slope-intercept form ($y = mx + b$):

$10x - 6y = -4 \implies -6y = -10x - 4 \implies y = \frac{10}{6}x + \frac{4}{6}$

Simplifying:

$y = \frac{5}{3}x + \frac{2}{3}$

The slope ($m_1$) of Line 1 is $\frac{5}{3}$.

Line 2: $y = -\frac{3}{5}x - 6$

This is already in slope-intercept form. The slope ($m_2$) is $-\frac{3}{5}$.

Line 3: $-3y = 5x + 4$

Rewrite this in slope-intercept form:

$-3y = 5x + 4 \implies y = -\frac{5}{3}x - \frac{4}{3}$

The slope ($m_3$) of Line 3 is $-\frac{5}{3}$.

2. Compare slopes to determine relationships between lines:

• Line 1 and Line 2: Slopes are $\frac{5}{3}$ and $-\frac{3}{5}$.

  $\frac{5}{3} \times -\frac{3}{5} = -1$

  These lines are perpendicular.

• Line 1 and Line 3: Slopes are $\frac{5}{3}$ and $-\frac{5}{3}$.

  $\frac{5}{3} \neq -\frac{5}{3}$

  These lines are neither parallel nor perpendicular.

• Line 2 and Line 3: Slopes are $-\frac{3}{5}$ and $-\frac{5}{3}$.

  $-\frac{3}{5} \neq -\frac{5}{3}$

  The product of slopes:

  $-\frac{3}{5} \times -\frac{5}{3} = 1$

  These lines are neither parallel nor perpendicular.

Conclusion:

1. Line 1 and Line 2: Perpendicular
2. Line 1 and Line 3: Neither
3. Line 2 and Line 3: Neither

Would you like more details on any of these steps, or do you have any questions?

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Related Questions:

1. How can you derive the equation of a line if you are given two points?
2. What are the properties of perpendicular lines in different coordinate systems?
3. How do you determine the distance between two parallel lines?
4. What is the significance of slope-intercept form in graphing linear equations?
5. How does changing the slope affect the orientation of a line on a graph?

Tip:

When finding relationships between lines, always convert to slope-intercept form to easily identify slopes.