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

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

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18 Jun 2011, 04:40
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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
[Reveal] Spoiler: OA

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Re: if xy+z=x(y+z) which of the following must be true? [#permalink]

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18 Jun 2011, 05:57
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WishMasterUA wrote:
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

xy+z=xy+xz
z=xz
xz-z=0
z(x-1)=0

Means;
either z=0
OR
x-1=0 i.e. x=1

Ans: "E"
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Re: if xy+z=x(y+z) which of the following must be true? [#permalink]

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05 Feb 2012, 21:28
WishMasterUA wrote:
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???

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Re: if xy+z=x(y+z) which of the following must be true? [#permalink]

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06 Feb 2012, 01:32
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rohitgoel15 wrote:
WishMasterUA wrote:
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.

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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Re: if xy+z=x(y+z) which of the following must be true? [#permalink]

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13 Aug 2013, 16:31
Bunuel wrote:
rohitgoel15 wrote:
WishMasterUA wrote:
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.

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.

Hi Bunuel,

I was wondering if substituting the answer options would be a good approach here? I attempted by substituting the values in each answer option and got both B/D to be valid for the equation in the question. Interestingly, if we simplify the equation we see that in both answer choices we have the value of Y, which is immaterial. It would be great if you can let me know what you think of approaching similar questions through their answer choices.

Thanks!
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Re: if xy+z=x(y+z) which of the following must be true? [#permalink]

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13 Jun 2014, 19:48
Bunuel wrote:
rohitgoel15 wrote:
WishMasterUA wrote:
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.

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.

Hi Bunuel,

-Two questions: I took the approach of plugging in answer choices and multiple answers satisfied the "must be true part". Why is that wrong?

-When you came down to xz=z, why did you take the next step of factoring out the z and equating the whole thing to zero, meaning, why did you get to the point of z(x-1)=0?

Thanks

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Re: if xy+z=x(y+z) which of the following must be true? [#permalink]

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14 Jun 2014, 01:45
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russ9 wrote:
Bunuel wrote:
rohitgoel15 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

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.

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.

Hi Bunuel,

-Two questions: I took the approach of plugging in answer choices and multiple answers satisfied the "must be true part". Why is that wrong?

-When you came down to xz=z, why did you take the next step of factoring out the z and equating the whole thing to zero, meaning, why did you get to the point of z(x-1)=0?

Thanks

1.
"MUST BE TRUE" questions:
These questions ask which of the following MUST be true, or which of the following is ALWAYS true for ALL valid sets of numbers you choose. Generally for such kind of questions if you can prove that a statement is NOT true for one particular valid set of numbers, it will mean that this statement is not always true and hence not a correct answer.

So, for "MUST BE TRUE" questions plug-in method is good to discard an option but not 100% sure thing to prove that an option is ALWAYS true.

As for "COULD BE TRUE" questions:
The questions asking which of the following COULD be true are different: if you can prove that a statement is true for one particular set of numbers, it will mean that this statement could be true and hence is a correct answer.

So, for "COULD BE TRUE" questions plug-in method is fine to prove that an option could be true. But here, if for some set of numbers you'll see that an option is not true, it won't mean that there does not exist some other set which will make this option true.

2.
To get what x and z could be from xz = z, we should do the way shown in my solution.

Hope it helps.
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If xy+z=x(y+z), which of the following must be true? [#permalink]

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09 Oct 2015, 21:11
MBAjunkie16 wrote:

Hi Bunuel,

I was wondering if substituting the answer options would be a good approach here? I attempted by substituting the values in each answer option and got both B/D to be valid for the equation in the question. Interestingly, if we simplify the equation we see that in both answer choices we have the value of Y, which is immaterial. It would be great if you can let me know what you think of approaching similar questions through their answer choices.

Thanks!

Hi Bunuel,

I have the same question with MBAjunkie16. What if I substituing the vales in each anwser and got B to be valid?

Thank you so much!
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Re: If xy+z=x(y+z), which of the following must be true? [#permalink]

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18 Jan 2016, 23:48
WishMasterUA 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

x=1 can be easily found out, but then you can you find out the other condition of y or z. It can have multiple answers

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

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23 Jan 2016, 09:14
xy + z = x(y + z)
Subtract xy from both side ==> z=xz
Remeber we cannot divide by a variable unless we know it is not = 0.
So 0=xz-z ==> 0=z(x-1) ==> z=0 OR x=1

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

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28 Jun 2016, 14:53
Hi everyone!
Why, in this case, the option X = 0 and Z = 0 wouldn't apply? I believe an "or" option is better than "and" (more flexible), but that's not the point here. If both X = 0 AND Z = 0, I get 0*0 - 0 = 0, which also holds (regardless of Y's value).

Thanks!!!

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

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29 Jun 2016, 00:25
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nramirez wrote:
Hi everyone!
Why, in this case, the option X = 0 and Z = 0 wouldn't apply? I believe an "or" option is better than "and" (more flexible), but that's not the point here. If both X = 0 AND Z = 0, I get 0*0 - 0 = 0, which also holds (regardless of Y's value).

Thanks!!!

It's explained several times above, so please go through the discussion. The question asks which of the following MUST be true not COULD be true. Option A (x=0 and y=0) is not necessarily true, thus not correct.

Hope it's clear.
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Re: If xy+z=x(y+z), which of the following must be true? [#permalink]

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29 Jun 2016, 03:53
Bunuel wrote:
nramirez wrote:
Hi everyone!
Why, in this case, the option X = 0 and Z = 0 wouldn't apply? I believe an "or" option is better than "and" (more flexible), but that's not the point here. If both X = 0 AND Z = 0, I get 0*0 - 0 = 0, which also holds (regardless of Y's value).

Thanks!!!

It's explained several times above, so please go through the discussion. The question asks which of the following MUST be true not COULD be true. Option A (x=0 and y=0) is not necessarily true, thus not correct.

Hope it's clear.

The A option I got in GMAT Prep is not X = 0 AND Y = 0, but X = 0 AND Z = 0. That's basically why I assume it's a possible option, and failed to see why it isn't.
Can you help me here?
Thanks!!

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

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29 Jun 2016, 04:17
Hi nramirez,

As stated by Bunuel, this is a must be true question. The only two possible cases are:
1) X has to be equal to 1
or
2) Z has to be equal to 0.

So when you plug the values into the equation, one of the two cases will always be satisfied.

You are saying that you got the option as X = 0 or Z = 0. Try plugging few values

Say, X = 1, Y = 1 and Z = 1
XY + Z = 1 + 1 = 2
X(Y + Z) = 1(1 + 1) = 2

So X = 0 is not a must be true condition.

Hope it helps.

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

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16 Apr 2017, 09:09
After getting z=xz , I cancelled out the z and got x=1. So naturally, B, D and E were valid options. A also gave me the equation z=xz which is exactly the same thing so I went with A. What is wrong in my approach?

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

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16 Apr 2017, 13:21
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Top Contributor
hannahkagalwala wrote:
After getting z=xz , I cancelled out the z and got x=1. So naturally, B, D and E were valid options. A also gave me the equation z=xz which is exactly the same thing so I went with A. What is wrong in my approach?

Once we know that z = xz, we cannot divide both sides by z, because it COULD be the case that z equals zero, and we should never divide by zero.

Here's why:
We know that (2)(0) = (1)(0)
However, if we try to divide by both sides by 0, we may believe that we get: 2 = 1, which of course is not true.

Cheers,
Brent
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Re: If xy+z=x(y+z), which of the following must be true? [#permalink]

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25 Jun 2017, 18:03
WishMasterUA 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 = xy + xz

z = xz

z-xz = 0

z(1-x) = 0

z = 0 or x = 1 so x-1 = 0.

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Re: If xy+z=x(y+z), which of the following must be true?   [#permalink] 25 Jun 2017, 18:03
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