https://file.aiquickdraw.com/mathbot/file/1726459654308lrfxbukw.jpg

The problem asks for the equation of a line that is **perpendicular** to the line \( y = -\frac{1}{3}x - 3 \) and passes through the point \( (3,9) \).

### Step-by-Step Solution:
1. **Find the slope of the perpendicular line**:
   The slope of the given line is \( -\frac{1}{3} \). For a line to be perpendicular, the slopes must be negative reciprocals. So, the slope of the perpendicular line will be:
   \[
   m = 3
   \]

2. **Use the point-slope form**:
   The point-slope form of a line is:
   \[
   y - y_1 = m(x - x_1)
   \]
   Here, \( m = 3 \) and the point \( (x_1, y_1) = (3, 9) \).

   Substituting the values:
   \[
   y - 9 = 3(x - 3)
   \]

3. **Simplify**:
   Expanding the equation:
   \[
   y - 9 = 3x - 9
   \]
   \[
   y = 3x
   \]

Thus, the equation of the line is:
\[
y = 3x
\]

Would you like further clarification or details on any part of this?

Here are five related questions for further exploration:
1. What is the significance of the negative reciprocal in perpendicular lines?
2. How do you determine the slope of a line given its equation?
3. How can you find the equation of a line parallel to a given line?
4. How do you convert between point-slope form and slope-intercept form of a line?
5. What are the properties of lines with slopes of 0 and undefined?

**Tip:** For perpendicular lines, always check that the product of their slopes equals \(-1\).

Find an equation for the line with the given properties. Perpendicular to the line y = -(1/3)x - 3; containing the point (3,9).

Learn how to find the equation of a line that is perpendicular to y = -(1/3)x - 3 and passes through the point (3, 9). This detailed solution covers concepts of slope, point-slope form, and perpendicular lines. Suitable for Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation