Question

What is an equation of the line that passes through the point (6,8) and is perpendicular to a line with equation y = 3x + 5?

a) y - 8 = 3(x - 6)
b) y - 8 = -(x - 6)
c) y + 8 = (x + 6)
d) y + 8 = -(x + 6)

Answer 1

The equation of a line perpendicular to the line y = 3x + 5 and passing through the point (6,8) is y - 8 = -1(x - 6), which corresponds to option b.

Step-by-step explanation:

The subject of this question is finding the equation of a line perpendicular to a given line and passing through a specified point. The original line has the equation y = 3x + 5, which means it has a slope (m) of 3. For a line to be perpendicular to this one, it must have a slope that is the negative reciprocal of 3. Therefore, the slope of the perpendicular line must be -1/3. Using the point-slope form, which is y - y1 = m(x - x1), where (x1, y1) is the point the line passes through, and m is the slope, we can substitute the slope and the point (6, 8) in:

y - 8 = (-1/3)(x - 6)

To match the answer choices, we must simplify this equation. Multiplying through by -1 to both terms in the parentheses, we get:

y - 8 = -(1/3)(x - 6).

However, none of the answer choices show a fraction in the slope. Because perpendicular lines have slopes that are negative reciprocals, and the original line's slope is 3, the slope of our line should be -1. Therefore, the correct equation that passes through the point (6,8) is:

y - 8 = -1(x - 6), which is option b.