The question asks you to write the equation of a line that passes through the point $(-6, 9)$ and is perpendicular to a line that passes through the points $(-2, 1)$ and $(6, 7)$. This equation must be in slope-intercept form.

Step-by-step solution:

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

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

   Substituting the given points $(-2, 1)$ and $(6, 7)$:

   $m = \frac{7 - 1}{6 - (-2)} = \frac{6}{8} = \frac{3}{4}$

   So, the slope of the given line is $m = \frac{3}{4}$.

2. Find the slope of the perpendicular line: The slope of a line perpendicular to another line is the negative reciprocal of the original slope. If the original slope is $m$, the perpendicular slope is $-\frac{1}{m}$.

   Since the slope of the given line is $\frac{3}{4}$, the slope of the perpendicular line is:

   $m_{\perp} = -\frac{4}{3}$

3. Use point-slope form to find the equation: Now, use the point-slope form of a line equation:

   $y - y_1 = m(x - x_1)$

   Where $m = -\frac{4}{3}$ and $(x_1, y_1)$ is $(-6, 9)$:

   $y - 9 = -\frac{4}{3}(x + 6)$

4. Simplify to slope-intercept form: Distribute the slope:

   $y - 9 = -\frac{4}{3}x - 8$

   Add 9 to both sides:

   $y = -\frac{4}{3}x + 1$

Final Equation:

The equation of the line is:

$y = -\frac{4}{3}x + 1$

Do you want more details on any part of this solution? Here are 5 related questions to extend your understanding:

1. How do you calculate the slope of a line passing through two points?
2. What is the significance of the negative reciprocal when finding perpendicular slopes?
3. Can you explain the point-slope form of a line and its uses?
4. How would the equation change if the line were parallel instead of perpendicular?
5. What would the graph of the perpendicular lines look like?

Tip: Always remember that perpendicular lines have slopes that multiply to give $-1$.