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Manager  Joined: 18 Sep 2009
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If xy+z=x(y+z), which of the following must be true?  [#permalink]

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29 00:00

Difficulty:   35% (medium)

Question Stats: 65% (01:17) correct 35% (01:36) wrong based on 735 sessions

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If xy+z=x(y+z), which of the following must be true?

A. x=0 and y=0
B. x=1 and y=1
C. y=1 and z=0
D. x=1 or y=0
E. x=1 or z=0

Originally posted by TomB on 23 Sep 2010, 13:44.
Last edited by Bunuel on 20 Mar 2012, 23:12, edited 3 times in total.
Corrected OA
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If xy+z=x(y+z), which of the following must be true?  [#permalink]

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TomB wrote:
If xy+z=x(y+z), which of the following must be true?
A. x=0 and y=0
B. x=1 and y=1
C. y=1 and z=0
D. x=1 or y=0
E. x=1 or z=0

$$xy+z=x(y+z)$$;

$$xy+z=xy+xz$$;

$$xy$$ cancels out: $$xz-z=0$$;

$$z(x-1)=0$$.

Either $$z=0$$ (in this case $$x$$ can take ANY value) OR $$x=1$$ (in this case $$z$$ can take ANY value).

To elaborate more:

As the expression with $$y$$ cancels out, we can say that given expression $$xy+z=x(y+z)$$ does not depend on value of $$y$$. Which means that $$y$$ can take ANY value. So all answer choices which specify the exact value of $$y$$ are wrong.

Next: "AND" in answer choices means that BOTH values must be true, but "OR" in answer choices means that EITHER value must be true.

Hope it helps.
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Thanks again Bunuel . I somehow always got stucked at option C Your explanation is awesome Intern  Joined: 02 Sep 2010
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Thanks Bunuel..The OA given is wrong its E thanks for proving it ..I was left wondering how answer could be D,Because I also got my answer as E
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Re: If xy + z = x(y + z)  [#permalink]

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Bunuel wrote:
TomB wrote:
xy+z=x(y+z). which of the following must be true?
a) x=0 and y=0
b) x=1 and y=1
c)y=1 and z=0
d)x=1 or y=0
e) x=1 or z=0

$$xy+z=x(y+z)$$ --> $$xy+z=xy+xz$$ --> $$xy$$ cancels out --> $$xz-z=0$$ --> $$z(x-1)=0$$ --> either $$z=0$$ (in this case $$x$$ can take ANY value) OR $$x=1$$ (in this case $$z$$ can take ANY value).

To elaborate more:
As the expression with $$y$$ cancels out, we can say that given expression $$xy+z=x(y+z)$$ does not depend on value of $$y$$. Which means that $$y$$ can take ANY value. So all answer choices which specify the exact value of $$y$$ are wrong.

Next: "AND" in answer choices means that BOTH values must be true, but "OR" in answer choices means that EITHER value must be true.

Hope it helps.

Yeah the explanation was very helpful..
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Joined: 28 Dec 2011
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Re: Which of the following must be true - Gmatprep  [#permalink]

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Hi there! I'm happy to help with this. So, the question:
If xy + z = x(y + z), then
(A) x = 0 and z = 0
(B) x = 1 and y = 1
(C) y = 1 and z = 0
(D) x = 1 or y = 0
(E) x = 1 or z = 0

So, as a very general rough approximation rule, when variables are multiplied, the mathematical consequences will far more often involve the word "or" than the word "and." That's just a rough-and-ready rule to use if all else is lost.

Let's look at this question. We have: xy + z = x(y + z)

Distribute on the right side: xy + z = xy + xz

Subtract xy from both sides: z = xz

So first of all, we get this funny equation with x & z in it, without y, so that means: there's no restriction on y, only restrictions on x & z.

In solving the equation z = xz, you may be tempted to divide by z, but WHOA THERE! Before you divide by a variable, you have to ask yourself the deeply heartfelt question: could this variable equal zero? If so, then dividing by it would break all mathematical laws, and Santa Claus would find out you've been bad. So, what do we do?

We'll look at two cases: (i) z = 0, and (ii) z =/ 0
(i) if z = 0, then both sides of the equation x = xz equal zero, and the equation is true. So, one possibility that makes the equation true is z = 0
(ii) if z is not equal to zero, then we can divide by z, and get 1 = x. That's the other solution that makes the equation true.

The solutions are x = 1 or z = 0, answer choice E.

Here's another practice question for you.

http://gmat.magoosh.com/questions/110

The question at that link should be followed by a video explanation.

Please let me know if you have any questions about what I've said. Good luck tomorrow!! Mike _________________
Mike McGarry
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Re: Which of the following must be true - Gmatprep  [#permalink]

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mikemcgarry has given an exhaustive explanation!

Simply put, it's basic algebra. Simplify the expression and equate to zero.
Watch out for the use of "and" and "or" in the answer choices.
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GMAT 1: 780 Q51 V48 GRE 1: Q800 V740 Re: Triple variable algebra problem  [#permalink]

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xy + z = x(y+z)
=> xy+z = xy + xz
=> z = xz
=> z=0 or x=1

Option (E)
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singhmaharaj wrote:
If xy + z = x(y+z) which of the following must be true

A. X=0 and z=0

B. X=1 and y=1

C. y=1 and z=0

D. X=1 and y=0

A. X=1 and z=0

xy + z = x(y+z)-----------> xy + z = xy+xz ---------> z = xz --------> x=1 and Z= 0 or 1
Note :- Here the value of y does not matter since xy = xy going to be zero for any value of y
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singhmaharaj wrote:
If xy + z = x(y+z) which of the following must be true

A. X=0 and z=0

B. X=1 and y=1

C. y=1 and z=0

D. X=1 and y=0

A. X=1 and z=0

xy + z = x(y+z)
=> xy + z = xy + xz
=> z = xz

The statement is true for both options A and E.
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rakeshd347 wrote:
singhmaharaj wrote:
If xy + z = x(y+z) which of the following must be true

A. X=0 and z=0

B. X=1 and y=1

C. y=1 and z=0

D. X=1 and y=0

E. X=1 and z=0

xy+z=xy+xz…cancel xy form both sides….xz=z….deduct z from both sides xz-z=0 or z(x-1)=0 so x=1 and z=0

z(x-1) = 0 .... either x-1 can be zero or z can be zero...

possible options can be z=0 and x-1 be anything or z can be anything and x-1 can be zero ....

Also the question didn't mention anything about x,y,z being positive
Intern  Joined: 12 Jan 2013
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Correct question is
If xy+z = x(y+z), which of the following must be true?

1. x=0 and z=0
2. x=1 and y=1
3. y=1 and z=0
4. x=1 or y=0
5. x=1 or z=0

Solution :
xy + z = xy + xz
z = xz i.e. z(x-1) = 0
so, z = 0 or x = 1 or both
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GRE 1: Q169 V154 Re: If xy+z=x(y+z), which of the following must be true?  [#permalink]

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TomB wrote:
If xy+z=x(y+z), which of the following must be true?

A. x=0 and y=0
B. x=1 and y=1
C. y=1 and z=0
D. x=1 or y=0
E. x=1 or z=0

Factorise and rule
here => xz=z
so z=0 or x=1
hence E
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Re: If xy+z=x(y+z), which of the following must be true?  [#permalink]

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_________________ Re: If xy+z=x(y+z), which of the following must be true?   [#permalink] 11 Nov 2019, 14:45
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