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Let's break this down.

You are asked to find the equation of a line that:

1. Passes through the point \( (2, -9) \)
2. Is perpendicular to the given line \( y = x + 6 \).

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### Step 1: Find the slope of the given line.
The equation \( y = x + 6 \) is in slope-intercept form \( y = mx + b \), where:
- \( m \) (the slope) is 1.

### Step 2: Find the slope of the perpendicular line.
If two lines are perpendicular, the product of their slopes is \( -1 \). Therefore, the slope of the perpendicular line is:
\[
m_{\text{perpendicular}} = -\frac{1}{1} = -1
\]

### Step 3: Use the point-slope form.
The equation of a line with slope \( m \) passing through a point \( (x_1, y_1) \) is:
\[
y - y_1 = m(x - x_1)
\]
Plugging in \( m = -1 \) and the point \( (2, -9) \):
\[
y - (-9) = -1(x - 2)
\]
\[
y + 9 = -x + 2
\]

### Step 4: Solve for \( y \) to get the equation in slope-intercept form.
\[
y = -x + 2 - 9
\]
\[
y = -x - 7
\]

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### Final Answer:
The equation of the line is:
\[
y = -x - 7
\]

Would you like more details or have any further questions?

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Here are 5 related questions you might explore next:
1. What is the general relationship between the slopes of parallel lines?
2. How do you find the equation of a line given two points?
3. What changes if the given line was horizontal or vertical?
4. How do you convert from standard form to slope-intercept form?
5. How do you determine if two lines are neither parallel nor perpendicular?

**Tip:** If you ever forget how to write a line equation, the point-slope form is a great starting point!

Write an equation of the line passing through the point (2, -9) that is perpendicular to the line y = x + 6.

Learn how to find the equation of a line that is perpendicular to y = x + 6 and passes through the point (2, -9). This problem involves using the point-slope form and understanding the relationship between slopes of perpendicular lines. Suitable for students in Grades 9-10.

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