Write the equation of a line that is perpendicular to 
\[y=3x-2\] and that passes through the point 
\[(-9,5)\].


To find the equation of a line that is perpendicular to \( y = 3x - 2 \) and passes through the point \( (-9, 5) \), we can follow these steps:

1. **Identify the slope of the given line**: The equation \( y = 3x - 2 \) is in slope-intercept form, \( y = mx + b \), where \( m \) is the slope. Here, the slope \( m = 3 \).

2. **Determine the perpendicular slope**: For two lines to be perpendicular, their slopes must be negative reciprocals. Therefore, the slope of the line we want is \( -\frac{1}{3} \).

3. **Use the point-slope form of a line**: With the slope \( -\frac{1}{3} \) and the point \( (-9, 5) \), we can use the point-slope formula:
   \[
   y - y_1 = m(x - x_1)
   \]
   Substituting \( m = -\frac{1}{3} \), \( x_1 = -9 \), and \( y_1 = 5 \):
   \[
   y - 5 = -\frac{1}{3}(x + 9)
   \]

4. **Simplify the equation**:
   \[
   y - 5 = -\frac{1}{3}x - 3
   \]
   \[
   y = -\frac{1}{3}x + 2
   \]

Thus, the equation of the line that is perpendicular to \( y = 3x - 2 \) and passes through \( (-9, 5) \) is:
\[
y = -\frac{1}{3}x + 2
\]

Would you like further details on any part of this solution?

Here are some additional questions to consider:
1. How do we verify that two lines are perpendicular?
2. What is the point-slope form of a line, and when is it useful?
3. Can you find a line parallel to \( y = 3x - 2 \) that passes through \( (-9, 5) \)?
4. What are the conditions for two lines to be parallel?
5. How can we rewrite the equation of a line in different forms?

**Tip:** Remember, the slopes of perpendicular lines are negative reciprocals of each other.

Write the equation of a line that is perpendicular to y=3x-2 and that passes through the point (-9,5).

Learn how to find the equation of a line that is perpendicular to y=3x-2 and passes through the point (-9, 5). This solution involves key algebraic concepts like slope-intercept form, point-slope form, and the relationship between perpendicular slopes. Suitable for students in Grades 8-10.

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