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(1) x^2 + y^2 = 25. Two unknowns, hence cannot get the value of x. Not sufficient. (2) xy = 12. Two unknowns, hence cannot get the value of x. Not sufficient.

(1)+(2) Now, we can go strictly algebraic way, though I'd suggest the following: x^2 + y^2 = 5^2 always makes me think about Pythagorean triple (3, 4, 5): 3^2+4^2=5^2. Now, 3 and 4 fit perfectly in xy = 12 too, so x can be 3 or 4. Moreover if we take the sign into consideration then there are even more options for (x, y): (3, 4); (4, 3); (-3, -4); (-4, -3). So, we have 4 possible values of x. Not sufficient.

x=? a) x^2 + y^2 = 25 x^2+ y^2 = 5^2 but we cannot deduce value of x ---NS

b) x*y = 12 Clearly not sufficient

a+b) x^2+y^2+2xy= (x+y)^2

25+24= (x+y)^2 7 = x+y need y to calculate X --NS E

From (x+y)^2=49 --> x+y=7 or x+y=-7. Also, the point is not that we need y to calculate x, but that we can get 4 different values of y from this system of equation, which give 4 different values of x.
_________________

Statement 1: \(x^2 + y^2 = 25\) We have one equation and two unknowns. Hence we cannot solve this equaiton.

INSUFFICIENT

Statement 2: xy = 12 Again, we have one equation and two unknowns. Hence we cannot solve this equaiton.

INSUFFICIENT

Statement 1 and Statement 2 combined: \(x^2 + y^2 = 25\) and xy = 12

Whenever we have something of the form \(x^2 +/- y^2\) , try to make it into perfect squares. We know that \((x+y)^2 = x^2 + y^2 +2xy\) So, we need the value of xy to make statement 1 a perfect square and we have the value of xy from statement 2.

Adding 2xy = 24 on both sides in statement 1, we have \(x^2 + y^2 +2xy = 49\) or \((x+y)^2 = 7^2\) Still we cannot solve for the value of x

(1) x^2 + y^2 = 25 --> If you remember pythagorean triplets (i.e. one being 3-4-5 triangle). Using this as an example, we have no way of knowing what value x takes on... +/- 3 or +/- 4. Thus, INSUFFICIENT

(2) xy = 12 --> We have no way of knowing what value x takes on. Thus, INSUFFICIENT

...so A, B, and D are out at this point

Trying (1) + (2) together, we still do not have any way of knowing with certainty the value of x. Thus, A,B,C,D are out.

(1) x^2 + y^2 = 25 x can take multiple values. Eg: x=3, y=4 OR x=4, y=3 OR x=-3, y=4.. and so on. x and y can both take positive as well as negative values also. Insufficient.

(2) xy = 12. Here also x can take multiple values. Eg: x=4, y=3 Or x=-4,y=-3. x and y can take mutliple values, though both x/y will have same sign. Insufficient.

Combining the two statements, x/y both have same sign, but still more than one cases possible. Eg, x=4,y=3 OR x=-4,y=-3 etc. Insufficient. Hence E answer