To find $k$, we need to use the fact that the slopes of two parallel lines must be equal.

Step 1: Calculate the slope of the line passing through $(5, 11)$ and $(-3, 7)$.

The slope formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

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

For the points $(5, 11)$ and $(-3, 7)$:

$m_1 = \frac{7 - 11}{-3 - 5} = \frac{-4}{-8} = \frac{1}{2}$

So, the slope of the first line is $\frac{1}{2}$.

Step 2: Set the slope of the second line equal to the first, since the lines are parallel.

Now, the slope of the line passing through $(-2, k)$ and $(2, 5)$ must also be $\frac{1}{2}$. Using the slope formula for these points:

$m_2 = \frac{5 - k}{2 - (-2)} = \frac{5 - k}{4}$

Since the slopes must be equal, we set $m_2$ equal to $m_1$:

$\frac{5 - k}{4} = \frac{1}{2}$

Step 3: Solve for $k$.

To solve for $k$, cross-multiply:

$2(5 - k) = 4(1)$

Simplify:

$10 - 2k = 4$

Now, solve for $k$:

$10 - 4 = 2k$

$6 = 2k$

$k = 3$

Final Answer:

The value of $k$ is $3$.

Would you like further details or have any questions?

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Here are 5 related questions to deepen your understanding:

1. How do you find the equation of a line given two points?
2. What happens to the slopes of two perpendicular lines?
3. How do you find the y-intercept once you have the slope and a point?
4. How do you determine if three points lie on the same line?
5. What is the geometric interpretation of a slope of zero?

Tip: For any two points, the slope formula is the ratio of the change in y-coordinates to the change in x-coordinates. This gives a precise measure of how steep a line is!