Final answer:
The equation of the line perpendicular to y = 9x - 4 and passing through (7, -6) is y = -1/9x - 47/9, which does not match any of the provided options, indicating a potential error in the question.
Step-by-step explanation:
The student's question involves finding the equation of a line that is perpendicular to a given line and passes through a specific point. Since the given line is y = 9x - 4, its slope (m) is 9. The slope of a perpendicular line is the negative reciprocal of the given line's slope, which is -1/9 in this case.
To find the equation of the perpendicular line, use the point-slope form, y - y1 = m(x - x1), where (x1, y1) is the point the line passes through, which in this scenario is (7, -6). Plugging in the values gives us y - (-6) = (-1/9)(x - 7), which simplifies to y + 6 = (-1/9)x + 7/9. To put this into y = mx + b form, subtract 6 from both sides resulting in y = (-1/9)x - 6 + 7/9. Simplify the constant terms (-6 + 7/9) to get the final equation of y = -1/9x - 47/9.
However, none of the provided options includes -47/9 as the y-intercept. Therefore, there might have been a typo or error in the available choices. It is crucial to communicate such discrepancies to the student and clarify the situation.