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.