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# If y/x ≠ 1 or 0, what is the value of y/z? (1) |x + y|=|x + z| (2) |x

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If y/x ≠ 1 or 0, what is the value of y/z? (1) |x + y|=|x + z| (2) |x  [#permalink]

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22 Jun 2017, 05:22
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59% (02:22) correct 41% (02:29) wrong based on 86 sessions

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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|

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Re: If y/x ≠ 1 or 0, what is the value of y/z? (1) |x + y|=|x + z| (2) |x  [#permalink]

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22 Jun 2017, 05:35
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hazelnut wrote:
If y/x ≠ 1 or 0, what is the value of y/z?

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

(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
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Re: If y/x ≠ 1 or 0, what is the value of y/z? (1) |x + y|=|x + z| (2) |x  [#permalink]

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22 Jun 2017, 11:46
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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Re: If y/x ≠ 1 or 0, what is the value of y/z? (1) |x + y|=|x + z| (2) |x  [#permalink]

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22 Jun 2017, 12:46
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Ans :C
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Re: If y/x ≠ 1 or 0, what is the value of y/z? (1) |x + y|=|x + z| (2) |x  [#permalink]

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23 Jun 2017, 02:50
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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Re: If y/x ≠ 1 or 0, what is the value of y/z? (1) |x + y|=|x + z| (2) |x  [#permalink]

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19 Sep 2017, 04:40
1
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.

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If y/x ≠ 1 or 0, what is the value of y/z? (1) |x + y|=|x + z| (2) |x  [#permalink]

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06 Dec 2017, 08:35
Hi

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

I hope this approach will be appreciated by getting kudos...

bkpolymers1617 wrote:
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.

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If y/x ≠ 1 or 0, what is the value of y/z? (1) |x + y|=|x + z| (2) |x &nbs [#permalink] 06 Dec 2017, 08:35
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