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 Post subject: GMAT Algebra (Data Sufficiency)Posted: Fri Feb 24, 2012 6:08 pm

Joined: Sun May 30, 2010 3:15 am
Posts: 424
What is the value of x + y?
(1) x² + y² = 5
(2) xy = 2

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(E) If we use statement (1) by itself, we can solve it for x. x² = 5 – y². This yields two possible solutions, x = -√(5 – y²) and x = √(5 – y²). The desired sum, x + y, can be y + √(5 – y²) and y – √(5 – y²). Therefore we do NOT have a definite value of the desired sum. Statement (1) by itself is NOT sufficient.

If we use statement (2) by itself, it implements that y can NOT be 0 and we can solve it for x. x = 2/y. The desired sum, x + y, is 2/y + y. We do NOT know the value of y. Therefore we do NOT have a definite value of the desired sum. Statement (2) by itself is NOT sufficient.

If we use the both statements together, let’s manipulate them a little. First, multiply the second one by 2 to get 2xy = 4. Then add the first one to it. x² + 2xy + y² = 5 + 4. In the left side we have a well known formula of the square of a sum.
(x + y)² = 9. It yields two solutions, x + y = 3 and x + y = -3. If you plug in x = 2 and y = 1 or x = -2 and y = -1, then you'll see that the both options are possible. Statements (1) and (2) taken together are NOT sufficient to answer the question. The correct answer is E.
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Here is the explanation: x² + y² = 5
OR ---> (x + y)² – 2xy =5
OR ---> (x + y)² – 5 = 2xy

We know from statement 2 that xy = 2. Therefore, 2xy = 4.
Now we can deduce that (x + y)² – 2xy = 5 (From statement 1 and 2).
Subsitute xy = 2 into the equation:
--> (x + y)² – 2(2) = 5
(x + y)² = 5 + 4
(x + y)² = 9
therefore x + y = 3

So option C is the answer.

What is you opinion?
Thanks.

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 Post subject: Re: t.1, qt. 20: algebra, squares, quadratic equationsPosted: Fri Feb 24, 2012 6:48 pm

Joined: Sun May 30, 2010 2:23 am
Posts: 498
Quote:
(x + y)² = 9
therefore x + y = 3
That's the key moment in your reasoning. (x + y)² = 9 yields two solutions, x + y = 3 and x + y = -3.
If you plug in x = 2 and y = 1 or x = -2 and y = -1, then you'll see that the both options are possible.

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 Post subject: Re: t.1, qt. 20: algebra, squares, quadratic equationsPosted: Sat Jul 07, 2012 8:46 am

Joined: Sun May 30, 2010 3:15 am
Posts: 424
(2) xy = 2 . Define x in terms of y: x = y/2 and substitute into the equation (1):
(y/2)² + y² = 5
y²/4 + y² = 5 multiply everything by 4
y² + 4y² = 20
5y² = 20
y² = 4 and y = 2

Substitute for y: 2x = 2, x = 1
The value of x + y = 1 + 2 = 3

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 Post subject: Re: t.1, qt. 20: algebra, squares, quadratic equationsPosted: Sat Jul 07, 2012 9:18 am

Joined: Sun May 30, 2010 2:23 am
Posts: 498
Quote:
y² = 4 and y = 2
Note that each equation like this one yields two solutions: y = 2 and y = -2 .

But more important here is the following:
questioner wrote:
(2) xy = 2 . Define x in terms of y: x = y/2
xy = 2 implies x = 2/y.
So the equation (1) becomes
(2/y)² + y² = 5
4/y² + y² = 5
4 + (y²)² = 5y²
(y²)² – 5y² + 4 = 0
(y² – 1)(y² – 4) = 0
y² = 1 or y² = 4
This yields the four possible solutions: -2, -1, 1, 2 .
When you calculate x, you'll see that x + y can be -3 or 3 .

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