Final answer:
To find the equation of the line perpendicular to the given line and passing through the given point, we need to determine the slope of the given line and then find the negative reciprocal of that slope. The correct answer is option D) 3x - 4y = 7.
Step-by-step explanation:
To find the equation of the line perpendicular to the given line and passing through the given point, we need to determine the slope of the given line and then find the negative reciprocal of that slope. The given line has an equation of 3x - 4y = 8. To determine the slope of this line, we need to rewrite the equation in slope-intercept form, which is y = mx + b, where m is the slope. Rearranging the equation, we get:
3x - 4y = 8
-4y = -3x + 8
y = (3/4)x - 2
From this equation, we can see that the slope of the given line is 3/4.
The negative reciprocal of 3/4 is -4/3, as the product of two perpendicular slopes is always -1. Now, we can use the point (-6,5) to find the equation of the line passing through this point with a slope of -4/3. Inserting the values into the point-slope form of a linear equation, y - y1 = m(x - x1), we get:
y - 5 = -4/3(x - (-6))
y - 5 = -4/3(x + 6)
y - 5 = -4/3x - 8
y = -4/3x - 3
This equation is in standard form, and the correct answer is option D) 3x - 4y = 7.