leekaru14 wrote:

If p and q are distinct lines in the xy coordinate system such that the equation for p is y= ax + b and the equation for q is y= cx + d, is ac = a²?

(1) d = b + 2

(2) For each point (x, y) on p, there is a corresponding point (x, y + k) on q for some constant x.

I read this question in Princeton Review. In answer part, they show this:

ac = a² so a = c. But if a = 2 and c = -2, for example? I can't get it.

I think statement 2 should read " for some constant k"

is ac = a²?

or is \(ac-a^2= 0\) --> a(c-a)= 0 --> either a = 0 or a = c

so we need to know if both the lines have same slope or is the slope of line p = 0

(1) d = b + 2 -> nothing about the slopes , just says line q has y intercept which is 2 greater than the y intercept of p, they could still be parallel and have the same slope or not be parallel and differ in slope.

also line y could have 0 slope or not , Hence insufficient

2) For each point (x, y) on p, there is a corresponding point (x, y + k) on q for some constant k.

let 2 points on P be (2, 3) and (5, 7 ) ( we can take any 2 points )

for each of these 2 points on P there will be 2 points on q such that ( 2, 5) and ( 5, 9 ) here taking k as 2

( x coordinates of points on q remains same as those points in line P and Y co ordinates are increased by a constant k )

now slope is (y2- y1)/ (x2- x1)

if we find the slopes of line P and q taking points according to statement 2 , as shown above then both lines will have same slope

hence a= c , sufficient

( Obviously k can vary , still both the lines will give same slope )