Is |x - z| = |y - z| ? (1) x = y (2) |x| - z = |y| - z

Question

Is |x - z| = |y - z| ?

(1) x = y
(2) |x| - z = |y| - z

Here is my problem. I solved this two ways and got two different answers. The first method was to square both sides and simplify and in doing so I got the right answer. The other way was to take the positive and negative cases of the stem in which case I got two separate solutions and the incorrect answer, i.e.

(x-z) = (y-z) OR
(x-z) = (z-y)

Can someone tell me why the second method wouldn't be used in this case?

Source: Total GMAT Math

(x-z) = (y-z) OR
(x-z) = (z-y)

Can someone tell me why the second method wouldn't be used in this case?

Source: Total GMAT Math

Zarrolou · Answer

Is |x-z| = |y-z|?

1) x=y 
So  |y-z| = |y-z| , sufficient.

2) |x|-z = |y|-z 
So  |x|=|y|  this could mean  y=x  (as above) or  y=-x . In the case y=x the answer is YES, in the other case (x=-y) you get  |-y-z| = |y-z|  and the answer could be NO, consider z=1 and y=2 for example; or YES (all zeros).
Not sufficient

I am sorry but I did not get what you did for the second statement... and there is no need t square the terms here. This could be solved more easily

WholeLottaLove · Answer

Sorry, I did a poor job explaining what I did wrong.

Is |x-z| = |y-z|?

1) x=y

|x-z| = |y-z|

So, I did this one of two ways:

The first way: squaring both sides:
|x-z| = |y-z|
(x-z)*(x-z) = (y-z)*(y-z)
x^2-2xz+z^2 = y^2-2yz+z^2
x^2-2xz=y^2-2yz
x(x-2z)=y(y-2z)
x=y
y(y-2z)=y(y-2z)

Sufficient

The second way was to take the positive and negative cases of |x-z| = |y-z| i.e.

x-z = y-z
x=y
y-z=y-z
OR
x-z=z-y
x=y
y-z=z-y

See my problem?

RnH · Answer

Your mistake was that you solved the absolutes first and the variables inside it later. You should've solved the variables first as you already know that x=y, for both the positive and the negative values of either of the two variables. Thus the difference between x and z will be the same as is between y and z.

This is a great Plug-in question btw.

This is a great Plug-in question btw.

Zarrolou · Answer

2) |x|-z = |y|-z

You do NOT know that y=x from statement 2. In your method you substitute x=y, but this is not what 2 says

y=+-x <== this is what is says, so

x-z = y-z  if  y=x 
 y-z=y-z  true
but if  y=-x  you get  x-z=-x-z  that could be true or not. Same thing for the other case  x-z=z-y . Hope it's clear

WholeLottaLove · Answer

Hmmm...I think you misunderstood me (or maybe I misunderstood you?) I wasn't refering to 2) above. I was saying that for 1) I solved two different ways. One way was to square |x-z|=|y-z| and the other way was to take the positive and negative case of |x-z|=|y-z| i.e. (x-z)=(y-z) OR (x-z)= -(y-z).

Zarrolou · Answer

Ok, now I got it.

So the first case you get  y-z=y-z  which is fine.

Then you analyze the case in which
 |x-z|=x-z  so  x>z  in this scenario
 |y-z|=z-y  so  y<z  in this scenario

So  x  is a number greater than  z , and  y  is a number lesser than  z  => they cannot be equal ( x
eq{y} ).
This is not a valid scenario to substitute  x=y .

The only case in which  y-z=z-y  is true is when the terms are all zeros (as I stated in a previous post).

Hope it's clear.

WholeLottaLove · Answer

Interesting. It's always been my understanding that we solve out the stem as much as possible before we move on to the two statements?

Bunuel · Answer

That's not always correct way of solving. Consider this question, for example.

WholeLottaLove · Answer

Is |x-z| = |y-z|?

2) |x|-z = |y|-z 
|x|=|y|
x=y
OR
x=-y

Positive
|x-z| = |y-z|
x-z = (y-z)
x-z = y-z

y-z = y-z
OR
-y=z = y-z

Negative
|x-z| = |y-z|
(x-z) = -(x-z)
x-z = z-x

y-z = z-y
OR
-y-z = z+y

Is that why it is insufficient?

Zarrolou · Answer

I think you are over complicating it. Refer here: is-x-z-y-z-155459.html#p1243251 and tell me if everything is clear There is no need to consider all possible cases

WholeLottaLove · Answer

I see what you are saying and I understand the solution. You are right, I am over complicating it...I am just trying to figure out how to go about this problem and when to use what methods for what problems.

Thanks!!!

shelrod007 · Answer

Is |x-z| = |y-z|?

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

I am weak in DS so trying to approach the question to solve them faster

Is |x-z| = |y-z|?

In English : Is the distance between x and z equal to the distance between y and z ?

Statement 1: Says x = y in other words x and y are essentially the same point . So for the stem question distance would be the same and hence information here is sufficient .

Statement 2: Simplifying 
|x|-z = |y|-z

|x| = |y| which would mean y = x or y = -x ?

Hence distances for x and z and y and z are same if y = x . 
They differ for x and z and y and z if y = -x .

Kinshook · Answer

Asked: Is |x-z| = |y-z|?

1) x=y
x-z = y-z
|x-z| = |y-z|
SUFFICIENT

2) |x|-z = |y|-z
|x| = |y|
Case 1: 
x= y; x-z = y-z; |x-z| = |y-z|
Case 2:
x = -y; x-z = -y-z; |x-z| = |y+z|
NOT SUFFICIENT

IMO A

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Is |x - z| = |y - z| ? (1) x = y (2) |x| - z = |y| - z

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