Find the slope of a line perpendicular to the line whose equation is 9, x, minus, 3, y, equals, 819x−3y=81. Fully simplify your answer.

To find the slope of a line perpendicular to the line given by the equation \(9x - 3y = 81\), we need to follow these steps:

1. **Rewrite the equation in slope-intercept form ( \(y = mx + b\) ) to identify the slope \(m\):**
   
   Start with the equation:
   \[
   9x - 3y = 81
   \]

2. **Isolate \(y\) on one side:**
   
   Subtract \(9x\) from both sides:
   \[
   -3y = -9x + 81
   \]

   Divide each term by \(-3\):
   \[
   y = 3x - 27
   \]

   Now, the equation is in slope-intercept form \(y = mx + b\) where \(m = 3\).

3. **Identify the slope of the given line:**
   
   The slope \(m\) of the given line \(9x - 3y = 81\) is \(3\).

4. **Determine the slope of the line perpendicular to the given line:**
   
   The slope of a line perpendicular to another line is the negative reciprocal of the original slope.

   The negative reciprocal of \(3\) is:
   \[
   -\frac{1}{3}
   \]

Thus, the slope of the line perpendicular to the line \(9x - 3y = 81\) is:
\[
-\frac{1}{3}
\]

Would you like more details or have any questions?

Here are some related questions to consider:

1. How do you find the slope of a line given two points?
2. What is the slope-intercept form of a line?
3. How do you determine if two lines are parallel?
4. What is the general form of a linear equation?
5. How do you convert a linear equation from standard form to slope-intercept form?
6. What is the point-slope form of a line?
7. How do you find the equation of a line given a point and a slope?
8. How do you determine if two lines are perpendicular?

**Tip:** Always double-check the negative reciprocal when finding the slope of a perpendicular line to avoid mistakes.

Learn how to find the slope of a line perpendicular to the equation 9x - 3y = 81 with a fully simplified solution. This problem involves understanding linear equations, slopes, and perpendicular lines. Suitable for high school students (Grades 9-12).

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