Final answer:
The equation of the line perpendicular to 4x + y = 3 and passing through (4,-3) is c) y = 4x - 19.
Step-by-step explanation:
The question asks for the equation of the line that is perpendicular to the given line 4x + y = 3 and passes through the point (4,-3). First, we need to find the slope of the given line. We can rearrange the equation to slope-intercept form, y = -4x + 3, which reveals that the slope is -4. The slope of the perpendicular line will be the negative reciprocal of -4, which is 1/4. Now we use the point-slope form of the equation of a line, which is y - y1 = m(x - x1), substituting the slope (m = 1/4) and the coordinates of the point (x1 = 4, y1 = -3) to get y + 3 = 1/4(x - 4). Simplifying this equation leads to y = 1/4x - 4 + 3, or y = 1/4x - 1. This equation must be compared against the options given, translating our slope of 1/4 into one of the options with a slope of 4 (the negative reciprocal). The correct answer is option (c) y = 4x - 19, where multiplying our found slope by -4 gives us option c's actual slope, confirming it is the perpendicular line.