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555-605 (Medium)|   Must or Could be True|            
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Abhi13545001
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If xy + z = x(y + z), which of the following must be true? 19 Nov 2021, 04:03
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Why option B is not right?
i substituted values, and found it to be okay.
I am doing something wrong, can somebody please explain?
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If xy + z = x(y + z), which of the following must be true? 19 Nov 2021, 04:21
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TomB
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
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Abhi13545001, we are looking for MUST be true.
Now x=1 and y=2 will give 1*2+z=1(2+z)……2+z=2+z
So y could be 1, 2 or anything till the time x=1.

So, let us look for must be true.
\(xy + z = x(y + z)………..xy+z=xy+xz………xz-z=0………z(x-1)=0\)

When would this be possible: when x=1 or z=0.


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If xy + z = x(y + z), which of the following must be true? 10 Dec 2021, 03:37
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Bunuel
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if xy+z=x(y+z) which of the following must be true?
1)X=0 AND Z=0
2)X=1 AND Y=0
3)Y=1 AND Z=0
4)X=1 OR Y=0
5)X=1 OR Z=0

I agree with the answer i.e. X=1 OR Z=0
But can the answer be also X=1 AND Z=0 (provided there was an option) as both of them taken together also satisfy the equation???

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

Notice that we are asked "which of the following MUST be true"?

\(xy+z=x(y+z)\) --> \(xy+z=xy+xz\) --> \(xy\) 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).

Addressing your questions: it's not necessary \(z=0\) and \(x=1\) both to be simultaneously true. "AND" in the answer choices means that BOTH values must be true, but "OR" in the answer choices means that EITHER value must be true.

Answer: E.

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.

Hope it helps.
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Dear Bunuel.. really appreciate your way of attacking the prob.
please correct me where did i go wrong - my reasoning was
-> xy +z = xy +xz.
The moment i looked at it, a solution popped up ; put x =y =1 --> there it is z=z. ( 10 sec and job done )
retrospection - should i think this way - that x=y=1 is one of the possible solutions. or not necessarily if this equation is to uphold , x and y have to be equal to 1.

phewww.. my state of mind is -: i understand but not able to absorb .
please help !
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If xy + z = x(y + z), which of the following must be true? 10 Dec 2021, 15:32
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TomB
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
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Given: xy + z = x(y + z)
Expand the right side: xy + z = xy + xz
Subtract xy from both sides: z = xz
Subtract xz from both sides: z - xz = 0
Factor: z(1 - x) = 0

So, either z = 0 or x = 1
Answer: E
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If xy + z = x(y + z), which of the following must be true? 21 Jun 2023, 11:05
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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



why cant we substitute the values of option B , C & E so that all the option will be right ?


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If xy + z = x(y + z), which of the following must be true? 22 Jun 2023, 01:52
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gmatpro15
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



why cant we substitute the values of option B , C & E so that all the option will be right ?


bb @bunnel KarishmaB
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The question is what is given and what is asked.

What is given: xy + z = x(y + z)

You are asked what this implies. What MUST BE TRUE?

xy + z = xy + xz
z(x - 1) = 0
So either z is 0 or x = 1.

Look at option (B):
B. x = 1 and y = 1
Is it necessary that x = 1 and y = 1?

What if x = 4 and y = 0? Then also xy + z = x(y + z) holds if z = 0.
This means that x = 1 and y = 1 is not necessarily true.

This is where the difference between necessary and sufficient conditions come into play. Option (B) is a sufficient condition, not necessary but we are asked for a necessary condition.

Not try to find some values for which xy + z = x(y + z) holds such that z is not 0 and x is not 1. We can't.

So you need to start with what is given and arrive at the option, not the other way around.
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If xy + z = x(y + z), which of the following must be true? 20 Aug 2025, 05:44
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