To find the equation of line B, which is perpendicular to line A and passes through point P (3, 5), we start by determining the slope of line A.

From the equation of line A given as $y = -3x + 8$, the slope ($m$) is $-3$. For a line to be perpendicular to another, its slope must be the negative reciprocal of the other line's slope. Thus, the slope of line B will be:

$m_B = -\frac{1}{m_A} = -\frac{1}{-3} = \frac{1}{3}$

With the slope of line B known as $\frac{1}{3}$, and using the point $(3, 5)$ through which line B passes, we can use the point-slope form of the equation of a line:

$y - y_1 = m(x - x_1)$

Plugging in the point and slope:

$y - 5 = \frac{1}{3}(x - 3)$

To convert this into the slope-intercept form ($y = mx + b$), we'll solve for $y$:

$y = \frac{1}{3}x - \frac{1}{3} + 5$

$y = \frac{1}{3}x + \frac{14}{3}$

Thus, the equation of line B is:

$y = \frac{1}{3}x + \frac{14}{3}$

Would you like to go over any step in more detail or do you have any questions about this?

Here are some related questions:

1. What would be the equation of a line parallel to line B?
2. How would the graph change if the slope of line A were positive?
3. Can you find the point of intersection between line A and line B?
4. What happens to the equation of line B if point P is changed?
5. How do you find the distance from a point to a line?

Tip: When checking if two lines are perpendicular, ensure that their slopes multiply to -1. This is a quick way to verify the relationship without needing a graph.