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To solve this problem, let's go through the necessary steps.

### Problem Analysis:
We need to find the equation of a line that is:
1. Perpendicular to the line \( y = \frac{1}{3}x + 12 \).
2. Passes through the point \( (2, 2) \).

### Step 1: Find the Slope of the Perpendicular Line
The given line has a slope of \( \frac{1}{3} \). For two lines to be perpendicular, their slopes must be negative reciprocals. So, the slope \( m \) of the line we need to find will be:
\[
m = -3
\]

### Step 2: Use the Point-Slope Form
With the slope \( m = -3 \) and the point \( (2, 2) \), we can use the point-slope form of a line:
\[
y - y_1 = m(x - x_1)
\]
Substitute \( m = -3 \), \( x_1 = 2 \), and \( y_1 = 2 \):
\[
y - 2 = -3(x - 2)
\]
Expanding this equation:
\[
y - 2 = -3x + 6
\]
\[
y = -3x + 8
\]

### Conclusion:
The equation of the line is:
\[
y = -3x + 8
\]

### Answer:
The correct answer is:
**\( y = -3x + 8 \)**

Would you like further clarification on any steps?

Here are 5 questions related to this concept:

1. How do you determine the slope of a perpendicular line given a slope?
2. What is the point-slope form of a line, and when is it used?
3. Why do perpendicular lines have slopes that are negative reciprocals?
4. How do you convert from point-slope form to slope-intercept form?
5. What happens to the equation if the line were parallel instead of perpendicular?

**Tip:** Always check that the calculated line passes through the given point by substituting the point back into the equation to verify correctness.

Write the equation of the line that is perpendicular to y = 1/3x + 12 and goes through the point (2, 2).

Learn how to find the equation of a line perpendicular to y = 1/3x + 12 that passes through the point (2, 2). This problem involves understanding slopes of perpendicular lines and using point-slope form. Suitable for students in Grades 8-10.

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