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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.

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.

Find the equation of a line that passes through (-6, 9) and is perpendicular to the line passing through (-2, 1) and (6, 7). Learn to use slope-intercept form and key algebraic formulas. Suitable for Grades 9-12.

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