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If x2 - y2 = 0 and xy 0, which of the following MUST be true

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If x2 - y2 = 0 and xy 0, which of the following MUST be true [#permalink] New post 20 May 2018, 10:49
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If \(x^2 - y^2 = 0\) and \(xy \neq 0\), which of the following MUST be true?
Indicate all such statements.

A. x = y
B. |x| = |y|
C. \(\frac{x^2}{y^2}=1\)
[Reveal] Spoiler: OA
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Re: If x2 - y2 = 0 and xy 0, which of the following MUST be true [#permalink] New post 23 May 2018, 13:40
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The answers are B and C.

Let's go through each answer one by one.

A. x=y
At first glance, this answer looks correct for the equation \(x^2 - y^2=0\). However, we have to keep in mind that any number, positive or negative, when squared, is positive. Because the only restriction is that \(xy > 0\), this means that either x or y could be positive while the other is a negative.

For example,

When x and y are both 5, then \(5^2 - 5^2 = 0\). However, if x=-5 and y=5, \((-5)^2 - 5^2 = 0\)

So A is false.

B. |x| = |y|
Using the last example, we can see that this is true. |-5| = |5|
The same goes for any combination of numbers.

B is true.

C. \(\frac{x^2}{y^2}=0\)
Following up from A, we already know that any number, positive or negative squared stays positive. If both numbers when squared, subtracted from each other are zero, then one number over the other will always be 1.

Example.
\(\frac{-5^2}{5^2}\) = 0
and \(\frac{5^2}{5^2}\) = 0

C is true.
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Re: If x2 - y2 = 0 and xy 0, which of the following MUST be true [#permalink] New post 26 May 2018, 12:59
Good one!
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If x2 – y2 = 0 and xy ≠ 0, which of the following must be tr [#permalink] New post 23 Jun 2018, 12:38
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If \(x^2\) – \(y^2\) = 0 and \(xy \neq 0\), which of the following must be true?

Indicate all such statements.

A. \(x = y\)
B. \(|x| = |y|\)
C. \(\frac{x^2}{y^2}= 1\)
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Sandy
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Joined: 07 Jun 2014
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Re: If x2 – y2 = 0 and xy ≠ 0, which of the following must be tr [#permalink] New post 18 Jul 2018, 07:10
Expert's post
Explanation

Since \(x^2 - y^2 = 0\), add \(y^2\) to both sides to get \(x^2 = y^2\). It might look as though x = y, but this is not necessarily the case. For example, x could be 2 and y could be –2.

Algebraically, taking the square root of both sides of x2 = y2 does not yield x = y, but rather |x| = |y|. Thus, the 1st statement is not necessarily true and the 2nd statement is true. The 3rd statement is also true and can be generated algebraically:

\(x^2 - y^2 = 0\)
\(x^2 = y^2\)
\(\frac{x^2}{y^2}= 1\).
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Sandy
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Re: If x2 - y2 = 0 and xy 0, which of the following MUST be true [#permalink] New post 23 Jul 2018, 03:12
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sandy wrote:
If \(x^2 - y^2 = 0\) and \(xy \neq 0\), which of the following MUST be true?
Indicate all such statements.

A. x = y
B. |x| = |y|
C. \(\frac{x^2}{y^2}=1\)



Given

\(x^2 - y^2 = 0\)

x^2 = y^2

Note: x and y could be negative or positive. Squared value is equal , not their individual value.

A. clearly out. x=2 , y = -2. Squared value is equal.

B. Squared value has no difference with absolute value. both show the positive answer.

C. True. both x^2 and y^2 have equal value. at the very beginning we got it.


So, B and c are true in all aspects.
Re: If x2 - y2 = 0 and xy 0, which of the following MUST be true   [#permalink] 23 Jul 2018, 03:12
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If x2 - y2 = 0 and xy 0, which of the following MUST be true

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