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

How did you get B? I got E... Bunuel would you be able to provide us explanation?

Let me try to answer.

You are given that l1: y=ax+b and l2: y=cx+d

Also, for ac=a^2 ---> a^2-ac=0 ---> a(a-c)=0 ---> either a=0 (line l1 is parallel to the y-axis) or a=c (slopes of lines l1 and l2 are equal). Thus for answering this question for "sufficiency", any statement or a combination that gives you either a=0 or a=c will be sufficient.

Per statement 1, d=b+2 ---> y-cx=y-ax+2 ---> x(a-c)=2. Not sufficient.

Per statement 1, what it means in simpler terms is that there is a point (x,y) on l1 such that for every single one of these points you have a corresponding point on l2 with coordinates (x,y+k) . In other words, if there is a point (m,n) on l1 then the corresponding point on l2 will be = (m,n+k), k being constant. Clearly, the 'x' coordinate of the 2 corresponding points on lines l1 and l2 is the same while the y coordinates differs just by a constant. Thus the slopes of lines l1 and l2 are equal ---> a=c.

You can think of it by drawing (l1) y=x+2 and (l2) y =x+4 follows what is mentioned in statement 2. If there is a point (1,3) on l1, you will get a corresponding (1,5=3+2) point on l2. This shows that the x-coordinates are the same , giving you the same slope of value 1.

Thus this statement is sufficient.

Hope this helps.

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

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.


When you modify the original condition, they become ac=c^2?, ac-c^2=0?, c(a-c)=0?, c=0 or a=c?. In the original condition, it says distinct lines, which becomes b=/d. That is, the question is if c=0? Or the slope of the two lines are the same, which is a=c?(are the two lines parallel?). In 2), it means l1 and l2 are parallel, which is yes and sufficient.
Thus, the answer is B.


 Once we modify the original condition and the question according to the variable approach method 1, we can solve approximately 30% of DS questions.

gmatophobia can you help me with this one

AnujL - We have two lines

\(l_1\)

\(y = ax + b\)

\(l_2\)

\(y = cx + d\)

Question: Is

\(ac = a^2\)

\(ac - a^2 = 0\)

\(a(c - a) = 0\)

This equality can hold true under two conditions

a = 0 → Inference: Line \(l_1\) is parallel to x axis

a = c → Inference: Line \(l_1\) is parallel to \(l_2\)

Target Question -

Is Line \(l_1\) is parallel to x axis

OR

Is Line \(l_1\) is parallel to \(l_2\)

Statement 1

(1) d = b + 2

This statement doesn't provide any information about either of the above two conditions. That is we can't infer whether line \(l_1\) is parallel to \(l_2\) or whether line \(l_1\) is parallel to the x-axis using this statement. Hence this statement alone is not sufficient.

Statement 2

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

Inference: For a given value of x, the difference between two y coordinates will always be constant. Hence, we can conclude that the lines are parallel.

This statement is sufficient as it helps conclude the second target question (Is Line \(l_1\) is parallel to \(l_2\))

Option B

gmatophobia can you elaborate the explanation for statement 2 with a suitable diagram ?

Can someone explain it with a diagram?
I marked B because I thought in statement 2, the x coordinate is the same for both lines and hence they have a point of intersection - hence not parallel (sufficient answer)
What concept am I misunderstanding here?