We are given two lines. Line $k$ passes through the points $(-3, -5)$ and $(1, 7)$. Line $h$ passes through the points $(-2, -10)$ and $(2, 2)$. We need to determine if line $k$ is parallel to line $h$, and explain why or why not.

1. Problem Description

We are given two lines. Line $k$ passes through the points $(-3, -5)$ and $(1, 7)$. Line $h$ passes through the points $(-2, -10)$ and $(2, 2)$. We need to determine if line $k$ is parallel to line $h$, and explain why or why not.

2. Solution Steps

Two lines are parallel if and only if their slopes are equal. Thus, we need to find the slope of each line and compare them.

The slope of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

For line $k$, the points are $(-3, -5)$ and $(1, 7)$. Let $(x_1, y_1) = (-3, -5)$ and $(x_2, y_2) = (1, 7)$.

Then the slope of line $k$ is:

$m_k = \frac{7 - (-5)}{1 - (-3)} = \frac{7+5}{1+3} = \frac{12}{4} = 3$

For line $h$, the points are $(-2, -10)$ and $(2, 2)$. Let $(x_1, y_1) = (-2, -10)$ and $(x_2, y_2) = (2, 2)$.

Then the slope of line $h$ is:

$m_h = \frac{2 - (-10)}{2 - (-2)} = \frac{2+10}{2+2} = \frac{12}{4} = 3$

Since $m_k = 3$ and $m_h = 3$, we have $m_k = m_h$. Therefore, the lines are parallel.

3. Final Answer

Yes. The lines have the same slope.