Determine whether the lines 

*L*1 and *L*2

 passing through the pairs of points are parallel, perpendicular, or neither.

*L*1: (−4, −3), (7, −3)

*L*2: (3, − 

1

7

), 4, − 

1

7

parallelperpendicular    neither

To determine whether the lines \( L_1 \) and \( L_2 \) are parallel, perpendicular, or neither, we first need to calculate their slopes.

### Step 1: Find the slope of \( L_1 \)
For \( L_1 \), passing through points \( (-4, -3) \) and \( (7, -3) \):

The slope formula is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substitute the values for \( L_1 \):

\[
m_1 = \frac{-3 - (-3)}{7 - (-4)} = \frac{-3 + 3}{7 + 4} = \frac{0}{11} = 0
\]

Since \( m_1 = 0 \), this indicates that \( L_1 \) is a horizontal line.

### Step 2: Find the slope of \( L_2 \)
For \( L_2 \), passing through points \( \left(3, -\frac{17}{7}\right) \) and \( \left(4, -\frac{17}{7}\right) \):

Substitute the values for \( L_2 \):

\[
m_2 = \frac{-\frac{17}{7} - \left(-\frac{17}{7}\right)}{4 - 3} = \frac{-\frac{17}{7} + \frac{17}{7}}{1} = \frac{0}{1} = 0
\]

Since \( m_2 = 0 \), this indicates that \( L_2 \) is also a horizontal line.

### Step 3: Compare the slopes

- **Parallel Lines**: The slopes are equal (\( m_1 = m_2 \)).
- **Perpendicular Lines**: The product of the slopes is \( -1 \) (\( m_1 \times m_2 = -1 \)).
- **Neither**: Neither condition is met.

Since both slopes are equal and \( m_1 = m_2 = 0 \), the lines \( L_1 \) and \( L_2 \) are **parallel**.

### Conclusion:
The lines \( L_1 \) and \( L_2 \) are **parallel**.

Would you like more details or have any questions?

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Here are 5 related questions for further practice:
1. How do you find the equation of a line parallel to a given line?
2. What are the slopes of vertical lines, and how does that affect their parallelism?
3. If you know the slope of a line, how do you determine a line perpendicular to it?
4. Can two distinct non-vertical lines ever have a slope of zero? Why or why not?
5. What is the general form of the equation for a horizontal line?

**Tip:** Horizontal lines always have a slope of zero, and they are parallel to each other if they lie on different y-values.

Explore how to determine if lines passing through specific point pairs are parallel, perpendicular, or neither. This detailed analysis covers the calculation of slopes and conditions for parallelism or perpendicularity. Suitable for Grades 9-12.

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