Is |x| = y + z?

Question

Is |x| = y + z?

(1) x + y = z
(2) x < 0

I know how to solve this by plugging in numbers and could arrive at the right answer. I want to learn how to solve this using algebra. When I tried with algebra I get answer as A. Can some please verify my steps and explain me what I am missing? Thank you.

1. -x = y +z ==> x+y=z (This is 1. and satisfies the question with different values of y and z for a negative x)
x=y+z ==> is not the same as given equation.
So A is sufficient.

2. Clearly insufficient.

gmatsaga · Answer

|x| = y + x (A) x = y + z AND/IF x > 0 (B) x = - ( y + z ) AND/IF x <0 I don't get Statement (1) Shouldn't it be x = - y - z ?? But this is insufficient because we have to have the condition that x > 0 Statement (2) x < 0 <-- insufficient (this is the condition we are looking for in statement (1)) Clearly, Statement (1) and Statement (2) = (C)

pgmat · Answer

Statement 1, x=-y-z is same as -x=y+z. and this implies x<0 and hence the statement should be sufficient. This is where I got lost.

gmatsaga · Answer

Yeah I also got this wrong the first time because I thought Statement (1) was already sufficient. However, we need Statement (2).

Observer · Answer

Did you copy this problem down correctly? -x=y+z =/=> x+y=z.

Observer · Answer

This is where you're incorrect. -x=y+z alone does not imply that x<0 without the condition that y+z >0.

Bunuel · Answer

The answer to this question is E, not C.

Consider below 2 cases:
 x=-1 ,  y=1  and  z=0  -->  |x|=1  and  y+z=1  --> answer YES;
 x=-1 ,  y=2  and  z=1  -->  |x|=1  and  y+z=3  --> answer NO.

I think you refer to the following question:

Is  |x|=y-z ?

Note that  y-z  must be  \geq{0} , because absolute value (in our case  |x| ) can not be negative.

Generally question asks whether  y-z\geq{0}   and  whether the difference between them equals to  |x| .

(1) x + y = z -->  -x=y-z  
if  x>0  -->  y-z  is negative --> no good for us;
if  x\leq{0}  -->  y-z  is positive --> good.
Two possible answers not sufficient;

(2)  x<0 
Not sufficient (we need to know value of y-z is equal or not to |x|)

(1)+(2) Sufficient.

Answer: C.

Hope it's clear.

badribaba1984 · Answer

Hi ,

Though it looks reasonable , I am not sure on what is wrong with this logic.

1. -x = y +z ==> x+y=z (This is 1. and satisfies the question with different values of y and z for a negative x)
x=y+z ==> is not the same as given equation.
So A is sufficient.

harshavmrg · Answer

given that x = y+z or -x=Y+z

so 1. x+Y = z..... not sufficient
2. it does not tell everything either... so not sufficient..

together.... -x+y = z

-x = -y+z, hence not same as given in the question, hence the answer to the main question is NO and with C option we are answering it.

EvaJager · Answer

(1)  x = y-z . Then  |x|=|y-z|  which is either  y-z  or  z-y. 
In order to have  y+z=y-z , necessarily  z=0  and also  y\geq0. 
In order to have  y+z=z-y , necessarily  y=0  and also  z\geq0. 
Obviously not sufficient.

(2) Clearly not sufficient.

(1) and (2) 
We are in the case  x=y-z<0  so  |x|=z-y . For  |x|=y+z  as seen above we need  y=0  and  z\geq0.  Neither condition is guaranteed.
Not sufficient.

Answer E

Injuin · Answer

Answer is E. Plugged in a bunch of numbers.

1) x=3, y=4, z=7 gives us the answer no, but x= -5, y=3, z=2 gives us yes. Insufficient

2) x<0 insufficient since y and z are unknown

Put them together, x=-5, y=3, z=2 gives us yes, but x=-5, y=-7, z=-12 gives us no. Insufficient.

peacewarriors · Answer

Hi Bunuel!
I have a doubt.
|x|= y+z gives us two equations..
X= y+ z or -x= y+z 
Statement one says x+y=z .. It is not possible to get the above mentioned statements with this equation.
Statement two says x<0 ..this should be sufficient to answer right?

Similarly. In the question |x|= y-z.
We can x= y-z and -x= y-z
Statement one by substituting we get -x= y-z so this should be suffice to right?

Bunuel · Answer

Not sure I understand your logic there... Number plugging proving that E is the answer: is-x-y-z-133977.html#p1094912 Algebraic approach proving the same: is-x-y-z-133977.html#p1114409 Hope this helps.

peacewarriors · Answer

In the question |x|= y-z.
We get two equations I.e.(removing the modulus . x= y-z and -x= y-z
Statement one says x+z =y
Therefore by substituting we get -x= y-z. so this should be suffice to answer right? 
As the question stem also has the same equation.

Posted from my mobile device

Bunuel · Answer

Since the correct answer is E, then this is obviously not right. If x\geq{0} , the questions asks: is x=y+z ? If x<{0} , the questions asks: is -x=y+z ? When we combine the statements, since it's given that x<0, the questions becomes: is -x=y+z . From (1) we have that -x=y-z , which is not sufficient to get whether -x=y+z . Consider the examples given here: https://gmatclub.com/forum/is-x-y-z-133977.html#p1094912

shakun · Answer

(I think the question could have been better stated.)

My answer is 'E'.

statement-1: Insufficient.
rewrite x+y=z as x=z-y. 
If (z=0 and y=0, thus z+y=0 and x=z-y=0) then |x|=z+y. But, if (z=2 and y=1, thus z+y=3 and x=z-y=1) then |x| 
eq z+y.

statement-2: Insufficient as it doesn't give any info on y and z. Example, x=-3, z=-1, y=-2, thus |x| 
eq z+y OR x=-3, z=1, y=2, thus |x|=z+y.

1+2: Insufficient.
Here, we know that, x=z-y and x<0,
when z=1 and y=2 then x = z-y=-1, and |x| 
eq =y+z
but when z=0, y=1 then x=z-y=-1, and |x| = y+z

texas · Answer

The way I approached this question was this:

since abs(x)>=0, the question is asking is y+z>=0

Statement 1 says that Y=Z-X. This means that Y+Z=2Z-X. We don't have specific values for Z or X so not sufficient

Statement 2 says x>0. This alone means nothing. Insufficient.

Together x>0 and the question is whether 2Z-X>0. We have no indication of the direction of Z, so insufficient together.

profileusername · Answer

For statement (1) we derive that

x+y = z

-x = y - z

When x is negative, y-z>0 and when x is positive, y-z<0 .. Knowing that y-z would be greater or less than 0 is insufficient to give us a similar indication for y+z ..

My question is whether this is a correct understanding of the information and can it be universally applied for two variables getting subtracted and added?

abhimahna · Answer

Hi  TheMastermind  ,

Yeah, this is another way to approach such questions.

Since you don't have any information about the sign of (y-z), you cannot conclude anything for y+z.

This is universally accepted concept and is not specific to this question.

Sarabjeets746 · Answer

Hi  KarishmaB  . Please help. I can't understand any of the solutions here. Thank you.

Is |x| = y + z?

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Is |x| = y + z?

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