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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.

Q1: A straight line has a gradient of -3/4 and passes through the point (32, 12). Work out the equation of the line.
Q2: The diagram below shows point P and Line A. Line B is perpendicular to line A and passes through point P. What is the equation of line B?

Learn how to find the equation of a line with a given slope and point, and how to calculate the equation of a line perpendicular to it. This step-by-step guide walks you through using the point-slope form and slope-intercept form to derive the final equations, including the slope of perpendicular lines and the relationships between them. Suitable for Grades 9-11.

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