Final answer:
To find the equation of the line perpendicular to 3x + y = 12 that passes through (4,0), the negative reciprocal of the original slope (-3) is used, which is 1/3. However, none of the provided answer choices have this slope, indicating a possible error in the question or answer choices.
Step-by-step explanation:
The question asks for the equation of a line that is perpendicular to the given line 3x + y = 12 and passes through the point (4,0). To find this line, we first need to determine the slope of the original line. We can rewrite the given equation in slope-intercept form (y = mx + b) by isolating y:
3x + y = 12
y = -3x + 12
The slope of this line is -3. The slope of a line perpendicular to this one would be the negative reciprocal, which is 1/3. The equation of a line with slope 1/3 passing through (4,0) can be given by the point-slope form of a linear equation, which is:
y - y1 = m(x - x1)
where (x1, y1) is the point the line passes through and m is the slope. Substituting the slope 1/3 and the point (4,0) into this formula gives:
y - 0 = 1/3(x - 4)
Multiplying both sides by 3 to get rid of the fraction:
3y = x - 4
Finally by adding 4 to both sides:
3y + 4 = x
Rewriting in slope-intercept form:
y = (1/3)x + (4/3)
None of the answer choices match this equation exactly. Therefore, the student may have made a typo in the original question or provided incorrect answer options. If we look at the choices provided, (A) y = -2x + 8 and (B) y = 2x - 8 cannot be correct since their slopes are -2 and 2 respectively, both of which are not the negative reciprocal of -3. Option (D) y = -x + 2 has a slope of -1 which also can't be correct. Option (C) y = x - 2 has a slope of 1, but this is not the negative reciprocal of -3 either. So, the student should review the question and ensure the given options are accurate.