x+y = ?

1) y = 2x -1 clearly INSUFFICIENT. two unknown one equation.
for x = 0, y = -1
for x = 1/2, y = 0
for x = 1, y = 1

2) y^2 = - |1-2x|
y^2 + |1-2x| = 0, As we know that square of a number is always +ve and absolute value is always positive too. sum of two +ve entities can be zero only when their individual value is zero

Hence, y = 0 and |1-2x| will only be zero when x = 1/2

x+y = 1/2+0 = 1/2 --- definite, SUFFICIENT.

Option B is the correct answer.
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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.

x+y =?

1) y=2x-1
so x+y
x+2x-1
3x-1 (ns)

2) y^2=-|1-2x|

as LHS will be positive always..

so RHS has to be positive

so, y^2 = -(1-2x)
y^2 = 2x-1
(NS)

1+2

Y=2X-1
Y^2= 2X-1
(2x-1)^2 = (2x-1)
x=1/2, 1
y=0,1

so ans, E

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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x + y = ?

(1) y=2x−1
(2) y2=−|1−2x|

I have a doubt. Here no information about x and y are given as they are integers. Is it correct to assume that
y2=−|1−2x| , here y^2 cannot be a negative number and therefore y^2= 0 and mark B as sufficient. Can't y be a complex number as no information is given that y is integer?

y2 = - negative number I assumed that y = complex number and marked choice B as insufficient since y can be any complex number based on x, in my
above answer.

MathExperts! Can you please help clear my doubt. Is it wrong to assume in GMAT , that a number can be complex number if no information is given about it.

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