To solve this problem, we need to:

1. Find the slope of the given line that passes through the points (-2, 1) and (6, 7).
2. Determine the perpendicular slope to this line.
3. Use the point-slope form of the equation with the perpendicular slope and the point (-6, 9).
4. Convert the equation to slope-intercept form.

Step 1: Find the slope of the given line

The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substitute the points (-2, 1) and (6, 7): $m = \frac{7 - 1}{6 - (-2)} = \frac{6}{8} = \frac{3}{4}$ So, the slope of the line is $\frac{3}{4}$.

Step 2: Find the perpendicular slope

The slope of a line perpendicular to another line is the negative reciprocal of the original slope. Therefore, the perpendicular slope is: $m_{\perp} = -\frac{4}{3}$

Step 3: Use the point-slope form

The point-slope form of a line equation is: $y - y_1 = m(x - x_1)$ Substitute $m = -\frac{4}{3}$ and the point (-6, 9): $y - 9 = -\frac{4}{3}(x + 6)$ 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 in slope-intercept form is: $y = -\frac{4}{3}x + 1$

Let me know if you'd like further clarification or more details on any step!

Related Questions:

1. How do you find the slope of a line between two points?
2. What is the geometric meaning of perpendicular slopes?
3. Can you explain the point-slope form and its components?
4. What would happen if the original line were horizontal or vertical?
5. How do you check if two lines are perpendicular by comparing their slopes?

Tip: Always simplify slopes and equations whenever possible for cleaner results!