What is the value of x + y?

Statement 1:x^2 + y^2 = 5. Insufficient
Statement 2:xy=2. Insufficient
Using both statements, (x+y)^2-2xy=5
(x+y)^2-4=5
(x+y)^2=9
x+y=+3 or -3
Answer E

800score Official Solution:

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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