x + y = ?
(1) y=2x−1
(2) y^2=−|1−2x|

As this is a two variable equation with two unknowns there should be two equations to solve it.
From eq1 alone we cant find x+y so using eq2 we should see wether x+y can be found or not
From eq2 for x<1/2 |1-2x| is positive so y^2=2x-1 this happens only when y=y^2 so Y=1 => X=1
for x>1/2 Y^2=1-2X and Y=2x-1 this happens when Y=-1 => X=0
for x=1/2 Y=0 so, by using the above two conditions we can determine x+y

x + y = ?

Taking choice 1
(1) y=2x−1
x, y can be anything. When x =0, y=-1 or when x=1,y=1.
Therefore this choice is not sufficient.

(2) y2=−|1−2x|

x, y can be anything. When x =-1, y=- (3^0.5) or when x=1/2,y=0.
Therefore this choice is not sufficient.

Combining choice 1 and choice 2.
y=2x−1 , y2=−|1−2x|
Squaring choice 1. y^2 = (2x-1)^2 and substituting in choice 2 in place of y^2.
(2x-1)^2 = - |1-2x|

|1-2x| can also be written as (1-2x)^2 which can also be written as ((-1)(2x-1))^2 which is (2x-1)^2.
Therefore (2x-1)^2 = - (2x-1)^2
(2x-1)^2 + (2x-1)^2 =0
2((2x-1)^2)=0
x=1/2.
Substituting x value in choice 1 y=2x-1 gives y=0.
Therefore x+y= 1/2.
So, both statements (1) and (2) TOGETHER are sufficient to answer the question and answer choice C.

MATH REVOLUTION OFFICIAL SOLUTION:

Since we have 2 variables in the original condition, we also need 2 equations. Since we need both 1) and 2), the correct answer is likely C. Using con 1) & 2) together, we get x=1/2, y=0. This is unique and sufficient. Therefore, the correct answer is C. However, since this is an “absolute value” problem, which is one of the key questions, we should apply Common Mistake Type 4(A).

In case of con 1), the values of x, y are not unique. Therefore, it is not sufficient.
In case of con 2), the only values that satisfy y^2+|1-2x|=0 are x=1/2, y=0. Therefore, it is unique and sufficient. The correct answer is B.

If C and B are both correct answers, B is would be the final correct answer.

Note 1 : a=b=0 is required to satisfy |a|+|b|=0 or a2+b2=0. Also, solving this type of question usually takes over 5 minutes during the actual exam. However, if you understand the relationship between Variable Approach Method and Common Mistake Types, you will be able to solve this type of question in just about 2 minutes.

Note 2 : For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
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DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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All numbers on the GMAT are by default real numbers. So, we do not consider complex/imaginary numbers on the GMAT.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics