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The question asks you to 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 \((6, 7)\). This equation must be in slope-intercept form.

### Step-by-step solution:

1. **Find the slope of the given line**:
   The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated by the formula:

   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]

   Substituting the given 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 \(m = \frac{3}{4}\).

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. If the original slope is \(m\), the perpendicular slope is \(-\frac{1}{m}\).

   Since the slope of the given line is \(\frac{3}{4}\), the slope of the perpendicular line is:

   \[
   m_{\perp} = -\frac{4}{3}
   \]

3. **Use point-slope form to find the equation**:
   Now, use the point-slope form of a line equation:

   \[
   y - y_1 = m(x - x_1)
   \]

   Where \(m = -\frac{4}{3}\) and \((x_1, y_1)\) is \((-6, 9)\):

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

4. **Simplify to slope-intercept form**:
   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 is:

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

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Do you want more details on any part of this solution? Here are 5 related questions to extend your understanding:

1. How do you calculate the slope of a line passing through two points?
2. What is the significance of the negative reciprocal when finding perpendicular slopes?
3. Can you explain the point-slope form of a line and its uses?
4. How would the equation change if the line were parallel instead of perpendicular?
5. What would the graph of the perpendicular lines look like?

**Tip**: Always remember that perpendicular lines have slopes that multiply to give \(-1\).

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 that passes through the point (-6, 9) and is perpendicular to a line passing through the points (-2, 1) and (6, 7). This solution covers key concepts such as slope, perpendicular slopes, and the slope-intercept form. Suitable for students in grades 9-11.

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