Which of the following equations represents a line that is perpendicular to $$ y=-3 x+6 $$ and passes through the point, $$ (3,2) $$ ? A. $$ y=-3 x+1 $$ B. $$ y=\frac{1}{3} x+1 $$ C. $$ y=-\frac{1}{3} x+1 $$ D. $$ y=\frac{1}{3} x+3 $$

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To determine which equation represents a line perpendicular to $$ y = -3x + 6 $$ and passes through the point $$ (3, 2) $$, follow these steps:

1. **Find the slope of the given line:**
   
   The given line is in the slope-intercept form $$ y = mx + b $$, where $$ m $$ is the slope. For $$ y = -3x + 6 $$, the slope $$ m_1 = -3 $$.

2. **Determine the slope of the perpendicular line:**
   
   Perpendicular lines have slopes that are negative reciprocals of each other. Therefore, the slope $$ m_2 $$ of the perpendicular line is:
   $$
   m_2 = -\frac{1}{m_1} = -\frac{1}{-3} = \frac{1}{3}
   $$

3. **Use the point-slope form to find the equation of the perpendicular line:**
   
   The point-slope form is:
   $$
   y - y_1 = m(x - x_1)
   $$
   Plugging in $$ m = \frac{1}{3} $$ and the point $$ (3, 2) $$:
   $$
   y - 2 = \frac{1}{3}(x - 3)
   $$
   Simplifying:
   $$
   y = \frac{1}{3}x - 1 + 2
   $$
   $$
   y = \frac{1}{3}x + 1
   $$
   
4. **Match with the given options:**
   
   The equation $$ y = \frac{1}{3}x + 1 $$ corresponds to option **B**.

**Answer:** B. $$ y=\frac{1}{3} x+1 $$
The correct answer is $$ y = \frac{1}{3}x + 1 $$, which is option B.

Which of the following equations represents a line that is perpendicular to \( y=-3 x+6 \) and passes through the point, \( (3,2) \) ? A. \( y=-3 x+1 \) B. \( y=\frac{1}{3} x+1 \) C. \( y=-\frac{1}{3} x+1 \) D. \( y=\frac{1}{3} x+3 \)

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