Final answer:
To find a perpendicular line’s equation, we determine the negative reciprocal of the original line's slope, use the point-slope form with the given point, and then convert to slope-intercept form. It appears that none of the answer choices provided match the correct equation, y = (2/3)x - 6.33.
Step-by-step explanation:
To find the equation of a line that is perpendicular to another line and passes through a certain point, you first need to determine the slope of the original line. Given the equation 3x + 2y = 8, let's find its slope by rearranging it into the slope-intercept form, which is y = mx + b, where m represents the slope and b the y-intercept.
First, solve the given equation for y:
3x + 2y = 8
2y = -3x + 8
y = (-3/2)x + 4
The slope of this line is -3/2. Since we are looking for a line that is perpendicular to this one, the slope of the desired line must be the negative reciprocal of -3/2, which is 2/3. Now, using the point-slope form of a line y - y1 = m(x - x1), where (x1, y1) is the point (2, -5) and m is the new slope 2/3, we get:
y - (-5) = (2/3)(x - 2).
Then, simplify and put it into the slope-intercept form:
y + 5 = (2/3)x - (4/3)
y = (2/3)x - (4/3) - 5
y = (2/3)x - (15/3) - (4/3)
y = (2/3)x - (19/3).
To make this look like the answer choices given, we should multiply through by 3 to clear the fraction:
3y = 2x - 19
y = (2/3)x - 19/3, which simplifies to y = (2/3)x - 6.33. Based on the provided answer choices, none of them match this equation. We may need to review the answers provided to ensure they are accurate, as it appears there may have been a mistake. None of the provided answer choices are correct.