let's hit statement (2) first, since that's easier.smclean23 wrote:Is x2 equal to xy?
(1) x2 – y2 = (x + 5)(y - 5)
(2) x = y
Answer is B.....HUH?
if x and y are equal, then you can substitute x for y anywhere you see either one.
therefore, xy = xx (because you can take out y and substitute x) = x^2.
sufficient.
before we hit the other statement, let's rephrase:
x^2 = xy would mean x^2 - xy = 0, which would mean x(x - y) = 0.
therefore, here's a rephrase: is x = 0 or x = y?
(note that statement (2) becomes absolutely trivial to answer with this rephrase)
also, another rephrase, which is weird-looking but, as it turns out, supremely convenient for statement 1: is x(x - y) = 0?
statement (1):
number picking may be the best way to go here. it's a bit tricky, but we can cherry-pick a couple of choices that make both sides 0.
x = y = 5: answer to question = 'yes'
x = -5, y = 5: answer to question = 'no'
insufficient.
if we don't want to pick numbers, let's play with the algebra and see where it goes.
we could factor x^2 - y^2 into (x - y)(x + y), as we are wont to do in the vast majority of cases involving that expression, but that's a dead end here because there aren't any common factors with the expression on the right side.
therefore, try expanding the right side:
x^2 - y^2 = xy - 5x + 5y - 25
when we rephrased the question, we ran into the expression x^2 - xy along the way. since both of those terms are in the above equation, let's isolate the same combo: add y^2 to both sides and subtract xy from both sides.
x^2 - xy = y^2 - 5x + 5y - 25
x(x - y) = y^2 - 5x + 5y - 25
if the right side is 0, then the answer to the question is 'yes'; if it's not 0, the answer is 'no'.
the right side can be 0 (for instance, if x = y = 5). also, it's possible for the right side not to be 0 (for instance, if x = -5 and y = 5).
insufficient.
answer = a
