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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!

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) in slope-intercept form.

Solve the problem of finding the equation of a line that passes through the point (-6, 9) and is perpendicular to the line connecting (-2, 1) and (6, 7). Learn how to use slope formulas and point-slope form to reach the solution in slope-intercept form.

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