if x = u^2 - v^2, y = 2uv, and z = u^2 + v2, and if x = 11

[DanielS773]

if x = u^2 - v^2, y = 2uv, and z = u^2 + v2, and if x = 11, what is the value of z?

1) y = 60
2) u = 6

can you walk me through this? I understand B but was not sure how A is sufficient.

[RonPurewal]

if x = u^2 - v^2, y = 2uv, and z = u^2 + v2, and if x = 11, what is the value of z?

1) y = 60
2) u = 6

can you walk me through this? I understand B but was not sure how A is sufficient.

well, the deal is that y + z factors out to (u + v)^2. see where you can get from there.

[RonPurewal]

but first...
is this problem actually in the GMAT Prep software?
the fact that you actually have to piece together a factoring formula makes me suspicious of the provenance of this problem.

if this problem really is from the GMAT Prep software, please post a screenshot-- no further discussion until screenshot. thanks.

it seems a bit too "mathy" to be an official problem.
i.e., it gives too much of an advantage to algebra whizzes; the official problems tend to be written in ways that don't require much, if any, knowledge of formulas.

[Kate_in_SS2]

Hello!

Dan emailed Student Services to let us know that he was having trouble posting the screenshot to the forum. We actually don't allow images to be attached or embedded into posts, for security reasons.

We've taken care of uploading the screenshot on Dan's behalf, and you can see it here:

http://postimg.org/image/kzrk8dla7/

Please let us know if you need anything else!

[DanielS773]

still not sure i understand. can you walk me through steps to get to each answer?

[tim]

still not sure i understand. can you walk me through steps to get to each answer?

Please be more clear about what you're asking. What do you mean by "each answer"? There is only one answer to this problem. And when you refer above to A or B being sufficient, those are answer choices, not things that are sufficient or insufficient. It may seem like a trivial point, but being very clear and precise about what is and is not going on in data sufficiency is crucial to understanding the big picture.

[RonPurewal]

here are a couple of approaches.

BRUTE FORCE METHOD for statement 1:
we have 2uv = 60.
solve for one of the variables (doesn’t matter which one). let’s solve for v:
v = 30/u

now substitute into u^2 - v^2 = 11 (remember, we’re given that x = 11):
u^2 - (30/u)^2 = 11
u^4 - 11(u^2) - 900 = 0
(u^2 - 36)(u^2 + 25) = 0
…so u^2 = 36, since u^2 = -25 is impossible.

substituting back into u^2 - v^2 = 11 gives v^2 = 25. since we have both u^2 and v^2, we have z.

[RonPurewal]

a method with more finesse for statement 1:

if we add z + y, that's u^2 + 2uv + v^2 = (u + v)^2.
if we subtract z - y, that's u^2 - 2uv + v^2 = (u - v)^2.

now multiply:
(z + y)(z - y) = (u + v)^2 • (u - v)^2
(z + 60)(z - 60) = ((u + v)(u - v))^2
z^2 - 3600 = (u^2 - v^2)^2
z^2 - 3600 = 11^2 (...since u^2 - v^2 = x)
... and so we have z^2 = some number.
this means we have the value of z, since it's the positive square root of that number. (z can't be negative, since it's the sum of two squares.)

[RonPurewal]

^^ by the way, note that the "more finesse" method really isn't much more efficient than the "brute force" method.

this is another excellent time to mention the 2 secrets to success on this exam:
1/ Do stuff
2/ Don't NOT do stuff

basically, if you manipulate these expressions with an at least somewhat definite sense of purpose (i.e., not totally random manipulations), whatever you're doing will probably work.

[RonPurewal]

oh, i forgot statement 2.

statement 2 is much less challenging.
you have u = 6 and x = 11, so 36 - v^2 = 11.
this gives you v^2, and then you can just add u^2 and v^2 to find z.

[NL]

wow, a tough question.
I’m likely to stop at this point:
uv =30
u^2-v^2 =11

It seems like either (u,v)= (6,5) or (-6, -5) is satisfying both equations. Any way to know that one pair is not satisfying without actually solving the “giant” equation?

(I know, you’re going to say: Don't NOT do stuff. But when I saw the giant equation, I thought I did something wrong or didn’t see an “easier” approach, which often happens in many quant questions)

[RonPurewal]

both of those pairs do, in fact, satisfy both equations (and, moreover, they also satisfy u^2 - v^2 = 11).

i'm not sure where you are going from there.

[RonPurewal]

do note that the goal of the problem is to find z, and that both of those pairs give the same value of z.

so, if you were thinking that you had to come up with unique values of u and v... nope.
in fact, if a DS problem asks for a combination of variables (like u^2 + v^2 here), then trying to find the individual variables will virtually ALWAYS give the wrong answer.

[RonPurewal]

on DS in general, if you try to find more information than is actually necessary, you'll get the wrong answer. you need to focus on the right goal.

e.g.,

• if the problem is "what is the tens digit of the integer N?" but you're trying to find the value of N... you're dead.
you should expect to find the tens digit WITHOUT finding the entire number.

• if the problem is "is this area greater than 48?" (see DS #155 in OG 13th) and you're trying to find a number for the area... you're dead.
you should expect to find whether the area is greater than 48, WITHOUT finding the area itself.

...and so on.

[RonPurewal]

this aspect of DS--its ability to punish people who try to find more information than they actually need--is one of the primary reasons why DS exists in the first place. (multiple-choice problems can't do this; if a multiple-choice problem presents a situation in which you can actually find u and v, then you can just find them.)