What is the value of xy?

[RonPurewal]

courtesy of a student:

from GMAT PREP

What is the value of xy?

1. y = x + 1

2. y = x^2 + 1

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it shouldn't take you too long to figure out that each of the statements, taken individually, is insufficient.
if you have good enough algebraic intuition, you can figure this out by just looking at the expressions and realizing that they aren't going to give you xy. otherwise, you can look at some numbers:

STATEMENT (1) if x = 1 and y = 2, then xy = 2; if x = 2 and y = 3, then xy = 6.
insufficient.

STATEMENT (2) if x = 1 and y = 2, then xy = 2; if x = 2 and y = 5, then xy = 10.
insufficient.

at this point, this becomes a standard SIMULTANEOUS EQUATIONS problem.
remember that there are two ways to solve SIMULTANEOUS EQUATIONS problems:
1) LINE THEM UP AND DO ARITHMETIC
2) SUBSTITUTION

this one will go either way.

line them up:
y = x^2 + 1
y = x + 1
--------------
0 = x^2 - x <-- subtract

substitution:
since both expressions are equal to y, just set them equal to each other:
x + 1 = x^2 + 1
x = x^2
0 = x^2 - x

so, either way, you have x^2 - x = 0.
factor --> x(x - 1) = 0
x = 0 or 1

notice that this does NOT YET mean that the data are insufficient. you must ascertain that THE DESIRED QUANTITY has two or more different values to say "insufficient".

if x = 0, then y = 1 (from either of the statements) --> xy = 0.
if x = 1, then y = 2 (from either of the statements) --> xy = 2.
so this really is INSUFFICIENT.

answer = (e)

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two notes:

* if the question had instead been "What is |x - 1| + |y - 1| ?" -- with EXACTLY the same statements -- then the answer would be (c) rather than (e). make sure you know why.

* you might actually discover the answer to this problem really quickly, by pure chance. when you're "plugging in" to the individual statements, it's entirely possible that you'll discover (x = 0, y = 1) and (x = 1, y = 2) -- for both statements. if you realize that these same numbers work for both statements -- and that they give two different answers to the prompt question -- then you'll immediately know that the answer is (e).

[NanoBotZ44]

Hi Ron,

* if the question had instead been "What is |x - 1| + |y - 1| ?" -- with EXACTLY the same statements -- then the answer would be (c) rather than (e).

Here is my attempt :

there are 2 sets of solutions : (X,Y) = (1,0) or (1,2)

1st set : (X,Y) = (1,0)
|x-1| + |y-1|
= |1-1| + |0-1| = 1

2nd set : (X,Y) = (1,2)
|x-1| + |y-1|
= |1-1| + |2-1| = 1

Since both sets give the same value, either set of values is acceptable. Please let me know if my approach is correct.

Regards,
Andy

[RonPurewal]

Hi Ron,

* if the question had instead been "What is |x - 1| + |y - 1| ?" -- with EXACTLY the same statements -- then the answer would be (c) rather than (e).

Here is my attempt :

there are 2 sets of solutions : (X,Y) = (1,0) or (1,2)

1st set : (X,Y) = (1,0)
|x-1| + |y-1|
= |1-1| + |0-1| = 1

2nd set : (X,Y) = (1,2)
|x-1| + |y-1|
= |1-1| + |2-1| = 1

Since both sets give the same value, either set of values is acceptable. Please let me know if my approach is correct.

Regards,
Andy

should be (0, 1) and (1, 2). (you switched the "0" and "1" in the first pair of coordinates.)

also, you wrote "either set of values is acceptable", which seems to call into question your general understanding of data sufficiency.
clearly you understand the math in the problem, but there's no such thing as an "acceptable set of values" in a problem like this one.
it's a DS problem. the point is whether you can get two different answers to the problem.
if you can actually get two different answers, then that's "insufficient"; if you can't, and there's only one answer you can get (as is the case here), then that's "sufficient".

if you actually understand that, and were just being careless with words here, then, excellent. on the other hand, if you were actually literally referring to an "acceptable set of values" (whatever that might mean to you here), then that's not a thing here.