Let's solve the problem step by step:

We need to write the equation of a line that:

1. Passes through the point (-6, 9).
2. Is perpendicular to a line passing through the points (-2, 1) and (6, 7).

Step 1: Find the slope of the given line.

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

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

Substituting the coordinates of the 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 $m = \frac{3}{4}$.

Step 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. The negative reciprocal of $\frac{3}{4}$ is:

$m_{\text{perpendicular}} = -\frac{4}{3}$

Step 3: Write the equation of the line.

We now know the slope of the required line is $-\frac{4}{3}$, and it passes through the point (-6, 9). Using the point-slope form of the equation of a line:

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

Substituting $m = -\frac{4}{3}$, $x_1 = -6$, and $y_1 = 9$:

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

Simplifying:

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

Distribute the slope:

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

Now, add 9 to both sides:

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

Final Answer:

The equation of the line in slope-intercept form is:

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

Would you like more details on this solution, or do you have any other questions?

Here are some related questions for further practice:

1. What is the slope of a line perpendicular to a line with slope 2?
2. How do you find the equation of a line parallel to a given line?
3. How would the equation change if the line passed through a different point, like (-4, 5)?
4. Can a vertical line be perpendicular to another line? What is its slope?
5. How would you graph the equation of this line?

Tip: Remember, when two lines are perpendicular, their slopes are negative reciprocals of each other.