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To solve this problem, let’s go through it step by step:

### Step 1: Find the Slope of the Given Line
The line passes through the points \((-2, 1)\) and \( (6, 7)\).

The slope formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the given points:
\[
m = \frac{7 - 1}{6 - (-2)} = \frac{6}{8} = \frac{3}{4}
\]

### Step 2: Find the Slope of the Perpendicular Line
Since we need a line that is perpendicular to this one, we take the negative reciprocal of the slope. The perpendicular slope, \(m_{\perp}\), is:
\[
m_{\perp} = -\frac{4}{3}
\]

### Step 3: Use Point-Slope Form to Find the Equation of the Line
We want the equation of the line that passes through the point \((-6, 9)\) with slope \( -\frac{4}{3}\).

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

### Step 4: Simplify to Slope-Intercept Form
Expand and simplify to get \(y\) in terms of \(x\):
\[
y - 9 = -\frac{4}{3}x - 8
\]
\[
y = -\frac{4}{3}x + 1
\]

### Final Answer
The equation of the line in slope-intercept form is:
\[
y = -\frac{4}{3}x + 1
\]

Would you like a detailed explanation on any of the steps?

### Related Questions:
1. How do you determine the slope between any two points?
2. Why is the slope of a perpendicular line the negative reciprocal?
3. What are the steps to convert from point-slope form to slope-intercept form?
4. How do you verify if two lines are perpendicular?
5. What are some practical applications of perpendicular lines in geometry?

### Tip:
Remember, for perpendicular lines, the product of their slopes is always \(-1\).

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.

Learn how to write the equation of a line passing through (-6, 9) that is perpendicular to a line through points (-2, 1) and (6, 7). This solution covers finding slopes, using point-slope form, and converting to slope-intercept form. Ideal for students in Grades 8-10.

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