Final answer:
The equation of the line which is perpendicular to the line 12 - 4y - 8x = 0 and passes through the point (4, -1) is y = -1/2x + 1.
Step-by-step explanation:
To find the equation of the line that passes through the point (4, -1) and is perpendicular to the given line, we first need to determine the slope of the original line. The given line's equation is 12 - 4y - 8x = 0, which can be rearranged to the slope-intercept form y = mx + b, where m is the slope. Rearranging the equation gives y = 2x - 3, implying the slope is 2. A line perpendicular to this would have a slope which is the negative reciprocal, hence -1/2.
Now using the point-slope form of the equation y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope of the new line, we plug in the values (4, -1) and slope -1/2 to get y - (-1) = -1/2(x - 4), which simplifies to y + 1 = -1/2x + 2 or y = -1/2x + 1 as the equation of the line.