If line L in the xy-coordinate plane has a positive slope, w

[Peiyilin09]

Hi, I have the following question to ask about a GMAT problem from CAT 4. I get the trying smart number method, but I think it takes too long.. is there any way to quickly just see that the first statement is true? What if they are asking for y intercept instead of the x-intercept..is 1) still true? Thanks!
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If line L in the xy-coordinate plane has a positive slope, what is the x-intercept of L ?

(1) There are different points (a, b) and (c, d) on line L such that ad = bc.

(2) There are constants m and n such that the points (m, n) and (-m, -n) are both on line L.

[tim]

First, the statements are always *true*; you need to take their truth and see if that helps you find an answer to the question at the top. I assume you mean to ask how we can tell that the statement is *sufficient* for that purpose. There is an algebraic solution on the CAT itself, but this may be a little more intuitive:

If ad = bc, that means b/a = d/c so a line segment coming from the origin will have the same slope to reach (a,b) as it does to reach (d,c). This means these points are collinear with the origin, so the line passes through (0,0)

[Peiyilin09]

Hi Tim, that is the exact explanation I was looking for, which is more intuitive. Thanks much!

[tim]

glad to hear it!

[sfbay]

is answer D?

2 points define a line. from there can can get slope. can get intercepts.

Is the fact that it goes through origin material?? positive slope and statements lead to that, but is not enough to know 2 points define line.

Also in statement 2 could m=n=0 ? since it says points m,n and -m, -n do they have to be different points??

[RonPurewal]

I get the trying smart number method, but I think it takes too long..

it shouldn't "take too long". you should be able to test 3-4 cases in under thirty seconds.

be careful with this sort of self-defeating thought! the point here is to find as many ways as possible to solve the problems, not to "rank" or "prioritize" certain methods over others.

[RonPurewal]

is answer D?

2 points define a line. from there can can get slope. can get intercepts.

with this thought process, you would not get choice (d), because neither of the statements gives 2 specific points - they both contain unknowns that can stand infinitely many different numbers.

more importantly -
if you are thinking like this, you will get almost every single DS problem wrong at higher levels.

in DS problems, you must focus on finding EXACTLY the information you're ASKED to find... because that's usually what happens: you can find exactly the information you need, and basically nothing more.

here are a couple of other examples:

Say a DS problem says "What is m + n?" --> If you are trying to find the individual values of m and n, you will almost certainly get this problem wrong.

Say a DS problem says "What is the tens digit of N?" --> If you are trying to find the actual value of N (i.e., all the digits), you will almost certainly get this problem wrong.

Say a DS problem says "What is the slope of line L?" --> If you are trying to find the equation of line L, you will almost certainly get this problem wrong.

here, too, if you think you have to find the actual equation of the line, you can't solve this problem. you have to realize that, in this problem, the x-intercept is the only thing that matters.

Is the fact that it goes through origin material??

the question says, "What is the x-intercept of the line?"
so, yes, if you know the line passes through (0, 0), then you know that the x-intercept is 0.

positive slope and statements lead to that, but is not enough to know 2 points define line.

... but you don't have 2 specific points. again, you have variables that can stand for an infinite variety of points.
so, there can be infinitely many different lines, for either statement. but they all have the same x-intercept, so the statements are sufficient nonetheless.

Also in statement 2 could m=n=0 ? since it says points m,n and -m, -n do they have to be different points??

it's possible that m = n = 0, but, in that case, the x-intercept would still be 0. so that doesn't change anything.

[anujsingh1984]

First, the statements are always *true*; you need to take their truth and see if that helps you find an answer to the question at the top. I assume you mean to ask how we can tell that the statement is *sufficient* for that purpose. There is an algebraic solution on the CAT itself, but this may be a little more intuitive:

If ad = bc, that means b/a = d/c so a line segment coming from the origin will have the same slope to reach (a,b) as it does to reach (d,c). This means these points are collinear with the origin, so the line passes through (0,0)

Could you please elaborate, how (a,b) and (c,d) are having same slope

[tim]

Can you please elaborate on what you mean by your question? It is not mathematically valid because (a,b) cannot have a slope on its own, and I think there may be a typo in your question as well.