If xy = – 6, what is the value of xy ( x + y )? (1) x – y =

[Jelie]

20. If xy = - 6, what is the value of xy ( x + y )?
(1) x - y = 5
(2) = 18

[RonPurewal]

20. If xy = - 6, what is the value of xy ( x + y )?
(1) x - y = 5
(2) = 18

whenever you have a problem like this, with a relationship stated in the problem statement, you MUST be aware that this relationship holds regardless of which statement(s) is/are taken to be true.

therefore, in a problem like this, each individual statement is actually a set of simultaneous equations: whatever equation is actually in the statement + the equation in the question prompt.

(1)
as mentioned above, this is really 2 simultaneous equations:
xy = -6
x - y = 5
this system is most easily solved by substitution: solve for x = y + 5, and then substitute into the other equation.
(y + 5)y = -6
y^2 + 5y = -6
y^2 + 5y + 6 = 0
(y + 3)(y + 2) = 0
y = -2 or y = -3
substituting back into either of the above equations gives:
y = -2 --> x = 3
y = -3 --> x = 2
these give different values for the target quantity xy(x + y), so, insufficient.

NOTE that you can also solve this system of equations by inspection (meaning that you can just stare at the equations, start conjuring up pairs of x and y that multiply to -6, and then test them to see whether any of them also subtract to 5).
solving by inspection isn't foolproof - for instance, it definitely won't help you come up with "weird" solutions involving irrational square roots - but there are very few such equations with "weird" solutions on the gmat. the vast majority of quadratics - indeed, every quadratic that i recall actually seeing, although i won't be so foolish as to make a general declaration - on the gmat have integer solutions.

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(2)
the actual statament here is missing; it's supposed to say xy^2 = 18.
as stated above, this is really two simultaneous equations:
xy^2 = 18
xy = -6
DIVIDING the equations (divide the left sides and divide the right sides) gives y = -3.
you can plug back in to find x, at which point you can plug x and y into the expression xy(x + y) to generate a unique value.
(no need to actually do this arithmetic - you can just know that you'll be able to do it, since it's data sufficiency)
therefore sufficient.

NOTE: you can also solve (2) by traditional substitution: solve for, say, x = -6/y, and plug it into the second equation to give (-6/y)y^2 = 18 --> -6y = 18 --> y = -3.

answer = (b)