If l1 and l2 are distinct lines in the xy coordinate system

[BTGmoderatorDC]

If l1 and l2 are distinct lines in the xy coordinate system such that the equation for l1 is y = ax + b and the equation for l2 is y = cx + d, is ac = a^2 ?

(1) d = b + 2
(2) For each point (x, y) on l1, there is a corresponding point (x, y + k) on l2 for some constant x.

OA B

Source: Princeton Review

[Ian Stewart]

I'm not sure why they ask "is a^2 = ac", which will be true when a=0 or when a=c, when we can prove the narrower fact that a=c using one of the Statements. Here, a and c are the slopes of the two lines, so if we can be sure the lines have the same slope, we'll have sufficient information.

Statement 1 just tells us the y-intercept of one line is 2 units higher than the y-intercept of the other. That tells us nothing about the slopes of the lines, so is not useful.

Statement 2 tells us that every point on the second line is exactly k units higher than a point on the first line. So the lines must have the same slope, and Statement 2 is sufficient. If one wanted to prove that algebraically, we know if (g, h) and (p, q) are two points on the first line, then we have the points (g, h+k) and (p, q+k) on the second line. The slope of the first line is "rise/run" = (q-h)/(p-g), and the slope of the second line is (q+k - (h + k))/(p - g) = (q-h)/(p-g), so the slopes are identical.