A line passes through (2/3,-10/3) and is perpendicular to the line 3x – 5y + 6 = 0. 
Find its equation.

To find the equation of the line that is perpendicular to \(3x - 5y + 6 = 0\) and passes through the point \((\frac{2}{3}, -\frac{10}{3})\), follow these steps:

1. **Find the slope of the given line \(3x - 5y + 6 = 0\):**

   Convert the given line equation to the slope-intercept form \(y = mx + b\):
   \[
   3x - 5y + 6 = 0 \implies -5y = -3x - 6 \implies y = \frac{3}{5}x + \frac{6}{5}
   \]
   Here, the slope \(m\) of the line is \(\frac{3}{5}\).

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

   The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope.
   \[
   m_{\perpendicular} = -\frac{1}{m} = -\frac{1}{\frac{3}{5}} = -\frac{5}{3}
   \]

3. **Use the point-slope form to find the equation of the perpendicular line:**

   The point-slope form of a line is given by \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line.
   Here, \((x_1, y_1) = \left(\frac{2}{3}, -\frac{10}{3}\right)\) and \(m = -\frac{5}{3}\).

   Substituting these values into the point-slope form:
   \[
   y - \left(-\frac{10}{3}\right) = -\frac{5}{3}\left(x - \frac{2}{3}\right)
   \]
   Simplify this equation:
   \[
   y + \frac{10}{3} = -\frac{5}{3}x + \frac{5}{9}
   \]
   \[
   y = -\frac{5}{3}x + \frac{5}{9} - \frac{10}{3}
   \]
   \[
   y = -\frac{5}{3}x + \frac{5}{9} - \frac{30}{9}
   \]
   \[
   y = -\frac{5}{3}x - \frac{25}{9}
   \]

Thus, the equation of the line that is perpendicular to \(3x - 5y + 6 = 0\) and passes through \((\frac{2}{3}, -\frac{10}{3})\) is:
\[
y = -\frac{5}{3}x - \frac{25}{9}
\]

Would you like further details or have any questions?

Here are some additional questions you might want to explore:

1. How do you convert between different forms of a linear equation?
2. What is the geometric significance of the slope-intercept form?
3. How do you find the distance between a point and a line?
4. What is the process for finding the intersection of two lines?
5. How do you determine if two lines are parallel?

**Tip:** When finding the slope of a line given in standard form \(Ax + By + C = 0\), you can quickly determine it by using the formula \(m = -\frac{A}{B}\).

Learn how to find the equation of a line perpendicular to 3x – 5y + 6 = 0 passing through the point (2/3, -10/3). This problem involves calculating slopes, using point-slope form, and understanding perpendicular lines. Suitable for students in Grades 9-12.

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