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If y/z ≠ 1 or 0, what is the value of y/z? (1) |x + y|=|x + z| (2) |x 22 Jun 2017, 05:22
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

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75% (hard)

Question Stats:

52% (02:23) correct 48% (02:18) wrong based on 272 sessions

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If y/z ≠ 1 or 0, what is the value of y/z?

(1) |x + y|=|x + z|
(2) |x - y|=|x - z|
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C
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If y/z ≠ 1 or 0, what is the value of y/z? (1) |x + y|=|x + z| (2) |x 22 Jun 2017, 05:35
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If y/x ≠ 1 or 0, what is the value of y/z?

(1) |x + y|=|x + z|
(2) |x - y|=|x - z|
Show more

(1) Squaring both sides
\((x^2 + y^2 + 2xy) = (x^2 + z^2 + 2xz)\)
\(y(y+2x) = z(z+2x)\)
\(y/z = (y+2x)/(z+2x)\)
Not sufficient

(2) Similarly
\(y/z = (y-2x)/(z-2x)\)
Not Sufficient

However on combining both
\((y-2x)/(z-2x) = (y+2x)/(z+2x)\)
\((y+2x)/(y-2x) = (z+2x)/(z-2x)\)
\(y/2x = z/2x\) (Using componendo & Dividendo)
\(y/z = 2x/2x = 1\)
Hence Sufficient
Answer C
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If y/z ≠ 1 or 0, what is the value of y/z? (1) |x + y|=|x + z| (2) |x 22 Jun 2017, 11:46
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Imo C
1)|x + y|=|x + z| take value of x=3 , y =1,z=-8 | x + y|=4
|x + z| =|4-8|=4 but z can take value as 1 also so insufficient
Similarly statement 2 is not sufficient
Taken together
|x + y|=|x + z|
|x - y|=|x - z|
They z and y have to have same values hence suffiecient
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If y/z ≠ 1 or 0, what is the value of y/z? (1) |x + y|=|x + z| (2) |x 22 Jun 2017, 12:46
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Ans :C
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If y/z ≠ 1 or 0, what is the value of y/z? (1) |x + y|=|x + z| (2) |x 23 Jun 2017, 02:50
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using both statements, y/z =1 if x ≠ 0
but why x ≠ 0 ?
as y/x ≠ 0 or 1, means y ≠ 0 and y ≠ x.
y/x can be undefined.
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If y/z ≠ 1 or 0, what is the value of y/z? (1) |x + y|=|x + z| (2) |x 19 Sep 2017, 04:40
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If y divided by z can’t equal 1, then y and z themselves are not equal. You can use logic to figure this out or you can manipulate the non-equation by multiplying both sides by z:

(1) INSUFFICIENT: Test some cases here. If x = 0, then |y| = |z|. Remember that y and z cannot be the same number! This would work, then, if y = 2 and z = –2 (or vice versa). In this case, y / z = –1. (As long as x and y are the same number but opposite in sign, you can choose any values you want, and the quotient will be –1.)
 
If, on the other hand, x = 1, then |1 + y| = |1 + z|. Solve for the positive version:
1 + y = 1 + z
y = z 
 
That is an illegal response, since y can’t equal z. Try the negative version:
1 + y = –(1 + z)
1 + y = –1 – z
y + z = –2
 
Pick two values that make this statement true. For example, if y = –3 and z = 1, then y / z = –3. There are at least two possible values for y / z, so this statement is insufficient.
 
(2) INSUFFICIENT: Test some cases again. If x = 0, then |–y| = |–z|. Remember again that y and z cannot be the same number! This would work, then, if y = 2 and z = –2 (or vice versa). In this case, y / z = –1.
 
If, on the other hand, x = 1, then |1 – y| = |1 – z|. Since solving for the negative version worked better last time, start with the negative version this time:
1 – y = –(1 – z)
1 – y = –1 + z
2 = y + z
 
Pick two values that make this statement true. For example, if if y = 3 and z = –1, then y / z = –3. There are at least two possible values for y / z, so this statement is insufficient.
 
(1) AND (2) SUFFICIENT: For each statement alone, testing x = 0 produced the same outcome, so at the least, y and z could be “opposites” (the same number but opposite signs) and y / z = –1. Are there other cases, though, that would work for both statements?
 
Take a look at the full versions of the two statements that didn’t produce the illegal outcome x = y; that is, use the negative version of each:
From statement (1) x + y = –(x + z) which becomes y + z = –2x
From statement (2): x – y = –(x – z) which becomes 2x = y + z
 
Notice anything? There are similar terms in those equations. Remember that the problem asks about y and z, so manipulate the first equation to drop the x terms:
(1) 2x = –y – z
(2) 2x = y + z
 
Set the two right-hand sides equal and simplify:
–y – z = y + z
0 = 2y + 2z
0 = y + z
 
This final equation proves that y and z have to be opposites: if y = 2, then z = –2; if y = 3, then z = –3; and so on. In any case, then, y / z = –1.
 
The correct answer is (C).
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If y/z ≠ 1 or 0, what is the value of y/z? (1) |x + y|=|x + z| (2) |x 06 Dec 2017, 08:35
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Hi

Please see my graphical approach to the question and provide feedback.



bkpolymers1617
If y divided by z can’t equal 1, then y and z themselves are not equal. You can use logic to figure this out or you can manipulate the non-equation by multiplying both sides by z:

(1) INSUFFICIENT: Test some cases here. If x = 0, then |y| = |z|. Remember that y and z cannot be the same number! This would work, then, if y = 2 and z = –2 (or vice versa). In this case, y / z = –1. (As long as x and y are the same number but opposite in sign, you can choose any values you want, and the quotient will be –1.)
 
If, on the other hand, x = 1, then |1 + y| = |1 + z|. Solve for the positive version:
1 + y = 1 + z
y = z 
 
That is an illegal response, since y can’t equal z. Try the negative version:
1 + y = –(1 + z)
1 + y = –1 – z
y + z = –2
 
Pick two values that make this statement true. For example, if y = –3 and z = 1, then y / z = –3. There are at least two possible values for y / z, so this statement is insufficient.
 
(2) INSUFFICIENT: Test some cases again. If x = 0, then |–y| = |–z|. Remember again that y and z cannot be the same number! This would work, then, if y = 2 and z = –2 (or vice versa). In this case, y / z = –1.
 
If, on the other hand, x = 1, then |1 – y| = |1 – z|. Since solving for the negative version worked better last time, start with the negative version this time:
1 – y = –(1 – z)
1 – y = –1 + z
2 = y + z
 
Pick two values that make this statement true. For example, if if y = 3 and z = –1, then y / z = –3. There are at least two possible values for y / z, so this statement is insufficient.
 
(1) AND (2) SUFFICIENT: For each statement alone, testing x = 0 produced the same outcome, so at the least, y and z could be “opposites” (the same number but opposite signs) and y / z = –1. Are there other cases, though, that would work for both statements?
 
Take a look at the full versions of the two statements that didn’t produce the illegal outcome x = y; that is, use the negative version of each:
From statement (1) x + y = –(x + z) which becomes y + z = –2x
From statement (2): x – y = –(x – z) which becomes 2x = y + z
 
Notice anything? There are similar terms in those equations. Remember that the problem asks about y and z, so manipulate the first equation to drop the x terms:
(1) 2x = –y – z
(2) 2x = y + z
 
Set the two right-hand sides equal and simplify:
–y – z = y + z
0 = 2y + 2z
0 = y + z
 
This final equation proves that y and z have to be opposites: if y = 2, then z = –2; if y = 3, then z = –3; and so on. In any case, then, y / z = –1.
 
The correct answer is (C).
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If y/z ≠ 1 or 0, what is the value of y/z? (1) |x + y|=|x + z| (2) |x 28 May 2019, 13:39
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Quote:
If \(\frac{y}{x} ≠ 1\) or 0, what is the value of \(\frac{y}{z}\)?

(1) \(|x + y|=|x + z|\)
(2) \(|x - y|=|x - z|\)
Show more

Official Explanation:
If y divided by z can’t equal 1, then y and z themselves are not equal. You can use logic to figure this out or you can manipulate
the non-equation by multiplying both sides by z:
\(\frac{y}{z}≠1\)
\(y≠z\)

(1) INSUFFICIENT: Test some cases here. If \(x = 0\), then \(|y| = |z|\). Remember that y and z cannot be the same number! This would
work, then, if \(y = 2\) and \(z = –2\) (or vice versa). In this case, \(\frac{y}{z}= –1\). (As long as x and y are the same number but opposite in sign,
you can choose any values you want, and the quotient will be –1.)
If, on the other hand, \(x = 1\), then \(|1 + y| = |1 + z|\). Solve for the positive version:
\(1 + y = 1 + z\)
\(y = z\)
That is an illegal response, since y can’t equal z. Try the negative version:
\(1 + y = –(1 + z)\)
\(1 + y = –1 – z\)
\(y + z = –2\)
Pick two values that make this statement true. For example, if \(y = –3\) and \(z = 1\), then \(\frac{y}{z} = –3\). There are at least two possible
values for \(\frac{y}{z}\), so this statement is insufficient.

(2) INSUFFICIENT: Test some cases again. If \(x = 0\), then \(|–y| = |–z|\). Remember again that y and z cannot be the same number! This
would work, then, if \(y = 2\) and \(z = –2\) (or vice versa). In this case, \(\frac{y}{z} = –1\).
If, on the other hand, \(x = 1\), then \(|1 – y| = |1 – z|\). Since solving for the negative version worked better last time, start with the
negative version this time:
\(1 – y = –(1 – z)\)
\(1 – y = –1 + z\)
\(2 = y + z\)
Pick two values that make this statement true. For example, if if \(y = 3\) and \(z = –1,\) then \(\frac{y}{z} = –3\). There are at least two possible
values for \(\frac{y}{z}\), so this statement is insufficient.

(1) AND (2) SUFFICIENT: For each statement alone, testing \(x = 0\) produced the same outcome, so at the least, y and z could be
“opposites” (the same number but opposite signs) and \(\frac{y}{z} = –1\). Are there other cases, though, that would work for both
statements?
Take a look at the full versions of the two statements that didn’t produce the illegal outcome \(x = y\); that is, use the negative
version of each:
From statement (1) \(x + y = –(x + z)\) which becomes \(y + z = –2x\)
From statement (2): \(x – y = –(x – z)\) which becomes \(2x = y + z\)
Notice anything? There are similar terms in those equations. Remember that the problem asks about y and z, so manipulate the
first equation to drop the x terms:
(1) \(2x = –y – z\)
(2) \(2x = y + z\)
Set the two right-hand sides equal and simplify:
\(–y – z = y + z\)
\(0 = 2y + 2z\)
\(0 = y + z\)
This final equation proves that y and z have to be opposites: if \(y = 2\), then \(z = –2\); if \(y = 3\), then \(z = –3\); and so on. In any case, then, \(\frac{y}{z} = –1.\)
The correct answer is
Show Spoiler
(C)
.
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If y/z ≠ 1 or 0, what is the value of y/z? (1) |x + y|=|x + z| (2) |x 17 Sep 2019, 05:50
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hazelnut
If y/x ≠ 1 or 0, what is the value of y/z?

(1) |x + y|=|x + z|
(2) |x - y|=|x - z|
Show more

y/x≠(1,0): y≠x and y,x≠0

(1) |x + y|=|x + z|: insufic.
manipulating: [1] x+y=x+z…y=z; [2] x+y=-x-z…2x+y+z=0;
testing: x,y=(2,1)…|x+y|=|2+1|=3; |x+z|=(3,-3)…|2+z|=(3,-3)…z=(1,-5);

(2) |x - y|=|x - z|: insufic.
manipulating: [1] x-y=x-z…-y=-z…y=z; [2] x-y=-x+z…2x-y-z=0;
testing: x,y=(2,1)…|x-y|=|2-1|=1; |x-z|=(1,-1)…|2-z|=(1-2=-1,-1-2=-3)…z=(-1,-3);

(1&2) |x + y|-|x + z|=|x - y|-|x - z| square both sides… 4xy=4xz…y=z, sufic.

Answer (C)
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If y/z ≠ 1 or 0, what is the value of y/z? (1) |x + y|=|x + z| (2) |x 07 Oct 2023, 00:42
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BillyZ
If y/z ≠ 1 or 0, what is the value of y/z?

(1) |x + y|=|x + z|
(2) |x - y|=|x - z|
Show more

Responding to a pm:

(1) |x + y|=|x + z|

Square both sides to get
\(x^2 + 2xy + y^2 = x^2 + 2xz + z^2\)
\(y^2 + 2xy = z^2 + 2xz\) ..... (i)

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

Square both sides to get
\(x^2 - 2xy + y^2 = x^2 - 2xz + z^2\)
\(y^2 - 2xy = z^2 - 2xz\) ....... (ii)

Using both, adding (i) and (ii),
\(2y^2 = 2z^2\)
So |y| = |z| (We do not get y = z when we take square root on both sides mind you. We get that their absolute values are equal)

Then either y = z or y = -z
Since we are given that y/z ≠ 1 then y/z = -1

Answer (C)

Check this post: https://anaprep.com/algebra-squares-and-square-roots/
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