Give the equation of the line passing through the point (8,−6)(8,-6) that is perpendicular to y=4/5x+6y=4/5x+6.

Write your answer in slope- intercept form.

To find the equation of the line that passes through the point \((8, -6)\) and is perpendicular to the line given by \(y = \frac{4}{5}x + 6\), follow these steps:

1. **Identify the slope of the given line**:  
   The slope of the line \(y = \frac{4}{5}x + 6\) is \(\frac{4}{5}\).

2. **Determine the slope of the perpendicular line**:  
   The slope of the line perpendicular to this one will be the negative reciprocal of \(\frac{4}{5}\).  
   The negative reciprocal of \(\frac{4}{5}\) is \(-\frac{5}{4}\).

3. **Use the point-slope form to find the equation**:  
   The point-slope form of a line is given by:
   \[
   y - y_1 = m(x - x_1)
   \]
   where \(m\) is the slope and \((x_1, y_1)\) is the point the line passes through.  
   Substituting \(m = -\frac{5}{4}\) and \((x_1, y_1) = (8, -6)\), we have:
   \[
   y - (-6) = -\frac{5}{4}(x - 8)
   \]
   Simplify this equation:
   \[
   y + 6 = -\frac{5}{4}x + 10
   \]
   Subtract 6 from both sides:
   \[
   y = -\frac{5}{4}x + 4
   \]

So, the equation of the line in slope-intercept form is:
\[
y = -\frac{5}{4}x + 4
\]

Would you like more details, or do you have any questions about this process?

Here are 5 related questions to explore further:
1. How do you find the slope of a line given two points?
2. What is the equation of a line parallel to \(y = \frac{4}{5}x + 6\) passing through a different point?
3. How would the equation change if the line were horizontal?
4. What are the characteristics of perpendicular lines on a coordinate plane?
5. How does changing the slope affect the steepness of a line?

**Tip**: When dealing with perpendicular lines, always remember that their slopes are negative reciprocals of each other!

Learn how to find the equation of a line passing through the point (8, -6) that is perpendicular to y = (4/5)x + 6. This problem involves using the point-slope form of a line and understanding perpendicular slopes. Suitable for high school students.

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