Final answer:
The equation of the line perpendicular to y = (1/4)x - 7 passing through the point (-2, -6) is y = -4x - 14, which is not listed among the choices given. If choosing the closest option, y = -4x - 5 would be the most appropriate.
Step-by-step explanation:
The student is asking for the equation of a line that is perpendicular to the given line y = (1/4)x - 7 and that passes through a specific point, which is (-2, -6).
To find a perpendicular line, first we determine the slope of the given line. For the line y = (1/4)x - 7, the slope is 1/4. A line that is perpendicular will have a slope that is the negative reciprocal of this slope, which is -4 (since -4 * 1/4 = -1).
Now, using the slope-intercept form y = mx + b, where m represents the slope and b the y-intercept, we can plug in the slope (-4) and the point (-2, -6) to find b.
y = mx + b
-6 = -4(-2) + b
-6 = 8 + b
b = -6 - 8
b = -14
So, the equation of the line is y = -4x - 14. However, this option is not listed among the choices given. This implies there may have been a typo in the question or the choices provided. If there is no typo, and the student needs to choose the closest option, y = -4x - 5 would be the most appropriate choice, as it correctly reflects the slope needed for perpendicularity.