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Algebra

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Source: — Data Sufficiency |
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by maihuna » Tue Jan 18, 2011 4:28 am
yes seems to me that oa should be C. adding the two will give 5 which will result in 2.5 as req value.
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by MAAJ » Tue Jan 18, 2011 3:47 pm
What is the value of x² + y² ?

(1) x² + y² = 2xy + 1

x² - 2xy + y² = 1
(x + y)² = 1
|x + y| = 1
(x + y) = 1 OR (x + y) = -1
This is not sufficient to answer


(2) x² + y² = 4 - 2xy

x² + 2xy + y² = 4
(x + y)² = 4
|x + y| = 2
(x + y) = 2 OR (x + y) = -2
This doesn't look like a typical GMAT question, as the two statements contradict each other. Still not sufficient.

(3) Combining the 2 staments we get 4 different equations, which could be solved for many different values so still combined statements are not sufficient.

IMO Correct Answer [spoiler](E)[/spoiler]

Note: Remember that you CAN'T combine exponential expressions when they are being added or subtracted. You can ONLY combine them when they are multiplied or divided. So x² + y² can't be simplified but if it were x² * y² then it could be manipulated to (xy)²
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by anshumishra » Tue Jan 18, 2011 5:12 pm
MAAJ wrote:What is the value of x² + y² ?

(1) x² + y² = 2xy + 1

x² - 2xy + y² = 1
(x + y)² = 1 should be (x - y)² = 1
|x + y| = 1
(x + y) = 1 OR (x + y) = -1
This is not sufficient to answer


(2) x² + y² = 4 - 2xy

x² + 2xy + y² = 4
(x + y)² = 4
|x + y| = 2
(x + y) = 2 OR (x + y) = -2
This doesn't look like a typical GMAT question, as the two statements contradict each other. Still not sufficient.

(3) Combining the 2 staments we get 4 different equations, which could be solved for many different values so still combined statements are not sufficient.

IMO Correct Answer [spoiler](E)[/spoiler]

Note: Remember that you CAN'T combine exponential expressions when they are being added or subtracted. You can ONLY combine them when they are multiplied or divided. So x² + y² can't be simplified but if it were x² * y² then it could be manipulated to (xy)²
Answer should be : C
just add the two equations : x² + y² = 2xy + 1 and x² + y² = 4 - 2xy

=> 2(x² + y²) = 5
=> (x² + y²) = 5/2 --- Sufficient

(Now, this is not what you need to do in this question and my intention is not to make it complex, just for educational purpose, here is how you can solve it using graph) :

I have plotted the graph to show that value of x² + y² is fixed for the 4 different points (x,y) satisfying these 2 equations .

x² + y² = 2xy +1
=> |x-y| = 1

x² + y² = 4 - 2xy
=> |x+y| = 2
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Thanks
Anshu

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by Rahul@gurome » Tue Jan 18, 2011 9:36 pm
MAAJ wrote:(3) Combining the 2 staments we get 4 different equations, which could be solved for many different values so still combined statements are not sufficient
Do not force yourself to agree with the OA.
The problem is asking for the value of (x² + y²), not of x or y or (x + y) or (x - y).
Solve all those four equations, you'll get (x, y) as (-1.5, -0.5), (-0.5, -1.5), (0.5, 1.5), and (1.5, 0.5). For all of the solutions (x² + y²) = 2.5

MAAJ wrote:Note: Remember that you CAN'T combine exponential expressions when they are being added or subtracted. You can ONLY combine them when they are multiplied or divided. So x² + y² can't be simplified but if it were x² * y² then it could be manipulated to (xy)²
Not really.
If x² = 4 and y² = 1, we can always say (x² + y²) = 5 or (x² - y²) = 3

Addition and subtraction are fundamental operations of mathematics. You can combine any numbers (whether they are exponential or logarithmic or irrational or even complex) by using them. Only restriction is the numbers should be of same dimension, i.e. addition/subtraction of $2 with 3 mile is not permitted.
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by MAAJ » Wed Jan 19, 2011 4:56 am
Thanks all for the replys, I made the mistake to try to solve for x or y and forgot about trying to solve for the combo :( :oops: A new lesson learned.

About combining exponential expression, I was refering to simplifying (x² + y²), but now that I read again he was talking about combining the 2 equations... (why didn't I see that? lol... guess needed more coffee) :oops: :mrgreen: my bad!
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