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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] New post 18 Jun 2011, 03:40
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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] New post 18 Jun 2011, 04:57
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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] New post 05 Feb 2012, 20:28
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] New post 06 Feb 2012, 00:32
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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.

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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Re: if xy+z=x(y+z) which of the following must be true? [#permalink] New post 13 Aug 2013, 15:31
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.

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.


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] New post 13 Jun 2014, 18:48
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.

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.


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] New post 14 Jun 2014, 00:45
Expert's post
1
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BOOKMARKED
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.

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.


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.
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis ; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) ; 12. Tricky questions from previous years.

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: If xy+z=x(y+z), which of the following must be true? [#permalink] New post 23 Jun 2015, 21:07
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Re: If xy+z=x(y+z), which of the following must be true?   [#permalink] 23 Jun 2015, 21:07
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