Write an equation of the line passing through point $P\left(-8,\ 0\right)$ that is perpendicular to the line $3x-5y\ =\ 6$ .

To find the equation of a line passing through $ P(-8, 0) $ and perpendicular to the line $ 3x - 5y = 6 $, follow these steps:

1. **Rewrite the given line in slope-intercept form** to find its slope.

   Starting with:
   \[
   3x - 5y = 6
   \]
   isolate $ y $:
   \[
   -5y = -3x + 6
   \]
   \[
   y = \frac{3}{5}x - \frac{6}{5}
   \]
   So, the slope of this line is $ \frac{3}{5} $.

2. **Determine the slope of the perpendicular line.**

   Since perpendicular lines have slopes that are negative reciprocals, the slope of the line we are looking for is:
   \[
   -\frac{5}{3}
   \]

3. **Write the equation of the line in point-slope form** using the point $ P(-8, 0) $ and the slope $ -\frac{5}{3} $:
   \[
   y - y_1 = m(x - x_1)
   \]
   Substituting $ m = -\frac{5}{3} $, $ x_1 = -8 $, and $ y_1 = 0 $:
   \[
   y - 0 = -\frac{5}{3}(x + 8)
   \]
   Simplifying:
   \[
   y = -\frac{5}{3}x - \frac{40}{3}
   \]

Thus, the equation of the line passing through $ P(-8, 0) $ and perpendicular to $ 3x - 5y = 6 $ is:
\[
y = -\frac{5}{3}x - \frac{40}{3}
\]

Would you like more details or have any questions?

---

1. How do we determine the slope of a line given its equation?
2. Why do perpendicular lines have slopes that are negative reciprocals?
3. How can we convert an equation from standard form to slope-intercept form?
4. What other forms can we use to write the equation of a line?
5. How would this process change if we were given a different point?

**Tip:** When finding perpendicular slopes, remember to flip the fraction and change the sign.

Write an equation of the line passing through point $P(-8, 0)$ that is perpendicular to the line $3x - 5y = 6$.

Learn how to find the equation of a line passing through P(-8, 0) that is perpendicular to the line 3x - 5y = 6. This solution includes concepts of slope, point-slope form, and perpendicular slopes. Suitable for students in Grades 9-11.

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