Consider the line y=8/9x+8. Find the equation of the line that is perpendicular to this line and passes through the point (-8,3)

To find the equation of the line that is perpendicular to the line \( y = \frac{8}{9}x + 8 \) and passes through the point \((-8, 3)\), follow these steps:

1. **Determine the slope of the original line**: The slope \( m \) of the line given by the equation \( y = \frac{8}{9}x + 8 \) is \( \frac{8}{9} \).

2. **Find the slope of the perpendicular line**: The slope of a line perpendicular to another is the negative reciprocal of the original slope. Thus:
   \[
   m_{\text{perpendicular}} = -\frac{1}{\frac{8}{9}} = -\frac{9}{8}
   \]

3. **Use the point-slope form of the line equation**: The point-slope form is given by:
   \[
   y - y_1 = m(x - x_1)
   \]
   where \( (x_1, y_1) \) is the point the line passes through, and \( m \) is the slope. Here, \( (x_1, y_1) = (-8, 3) \) and \( m = -\frac{9}{8} \).

   Plugging in the values:
   \[
   y - 3 = -\frac{9}{8}(x + 8)
   \]

4. **Simplify the equation**:
   \[
   y - 3 = -\frac{9}{8}x - 9
   \]
   \[
   y = -\frac{9}{8}x - 6
   \]

Thus, the equation of the line that is perpendicular to \( y = \frac{8}{9}x + 8 \) and passes through the point \((-8, 3)\) is:
\[
y = -\frac{9}{8}x - 6
\]

### Summary
- The perpendicular line's equation is \( y = -\frac{9}{8}x - 6 \).

Do you want more details on any part of the process? Here are some related questions to consider:

1. What are the geometric interpretations of slopes and perpendicular lines?
2. How can we convert this equation into standard form?
3. What would be the intersection point of this line with another line?
4. Can you find the distance between the two lines?
5. How do you determine whether two lines are parallel or perpendicular using their slopes?

**Tip:** Always check the slopes when working with lines to confirm if they are parallel or perpendicular.

Consider the line y=8/9x+8. Find the equation of the line that is perpendicular to this line and passes through the point (-8,3).

Explore how to find the equation of a line perpendicular to y=8/9x+8 that passes through (-8,3). This solution highlights the concepts of slope, point-slope form, and perpendicular lines, suitable for Grades 8-10 students.

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