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

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sandy
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If x2 - y2 = 0 and xy 0, which of the following MUST be true [#permalink] New post   Updated on: 30 Nov 2024, 02:33
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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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B,C

Originally posted by sandy on 20 May 2018, 10:49.
Last edited by Carcass on 30 Nov 2024, 02:33, edited 1 time in total.
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PIneappleBoy2
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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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Re: If x2 - y2 = 0 and xy 0, which of the following MUST be true [#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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Re: If x2 - y2 = 0 and xy 0, which of the following MUST be true [#permalink] New post   18 Jul 2018, 07:10
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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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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.
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Re: If x2 - y2 = 0 and xy 0, which of the following MUST be true [#permalink] New post   04 Jul 2019, 10:49
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. \(\frac{x^2}{y^2}=1\)


Why We can't take a different value for x and y.? Like: x=2 and y=3, if we so then B. must not be an answer. Only answer C.

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Re: If x2 - y2 = 0 and xy 0, which of the following MUST be true [#permalink] New post   06 Jul 2019, 14:28
huda wrote:
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. \(\frac{x^2}{y^2}=1\)


Why We can't take a different value for x and y.? Like: x=2 and y=3, if we so then B. must not be an answer. Only answer C.




If you take different values for x andy, then will their square difference yields 0? Certainly not!!
?
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Re: If x2 - y2 = 0 and xy 0, which of the following MUST be true [#permalink] New post   06 Jul 2019, 14:33
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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. \(\frac{x^2}{y^2}=1\)


Let's think this way...
x^2 - y^2 =0
=> (x+y) (x-y)=0
=> x=y or x = -y
So we can eliminate option A. But their absolute value must be equal. We must take B as an answer.

Again,
x^2 - y^2 =0
=> x^2 = y^2
x^2/y^2=1

Definitely C is also our answer! So B,C.
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Re: If x2 - y2 = 0 and xy 0, which of the following MUST be true [#permalink] New post   08 May 2020, 04:55
PIneappleBoy2 wrote:
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.


You accidentally set C. equal to zero instead of 1. Might wanna fix that just so someone on the fence can follow. Thanks!
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Re: If x2 - y2 = 0 and xy 0, which of the following MUST be true [#permalink] New post   08 Oct 2022, 02:59
Can someone help with this:

If x^2 = Y^2 which of the following must be true?
1. x=y
2. x^2 - y^2 = 0
3. |x| - |y| = 0
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Re: If x2 - y2 = 0 and xy 0, which of the following MUST be true [#permalink] New post   08 Oct 2022, 03:23
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manvijain08 wrote:
Can someone help with this:

If x^2 = Y^2 which of the following must be true?
1. x=y
2. x^2 - y^2 = 0
3. |x| - |y| = 0


If this is a complete new question please follow the rules for posting

Open a new discussion under the right sub forum : PS under PS and so on and we will help. For example here https://gre.myprepclub.com/forum/multip ... choice-23/

As the question is stated above we do know many things

regards
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Re: If x2 - y2 = 0 and xy 0, which of the following MUST be true [#permalink] New post   30 Nov 2024, 02:33
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Re: If x2 - y2 = 0 and xy 0, which of the following MUST be true [#permalink]
30 Nov 2024, 02:33
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