What values of x have a corresponding value of y that satisfies both x

Question

What values of x have a corresponding value of y that satisfies both xy > 0 and xy = x + y ? A. x \leq -1 B. -1 < x \leq 0 C. 0 < x \leq 1 D. x > 1 E. All real numbers PS22680.02

BrentGMATPrepNow · Accepted Answer

Given:  xy = x + y 
Subtract  y  from both sides to get:  xy - y= x 
Factor:  y(x-1)=x 
Divide both sides by  x-1  to get:  y=\frac{x}{x-1}

Important: At this point, there are at least two different approaches we can take

APPROACH #1  : 
Now that we know  y=\frac{x}{x-1} , we can readily see that  x cannot equal 1 , otherwise, the denominator in y is 0, which would make y undefined.
At this point we can eliminate answer choices A, C and E, since they all allow for x to equal 1. 
Since we're down to just two answer choices, let's just test some x-value.

For example, if  x = 2 , then we get:  y=\frac{x}{x-1}=\frac{2}{2-1}=2 . 
So,  x = 2  and  y = 2  is a possible solution to the system of equations.
Since it's possible for x to equal 2, we can eliminate answer choice B

By the process of elimination, the correct answer is D.

APPROACH #2  : 
We've already concluded that:  y=\frac{x}{x-1}

Now take the given inequality,  xy > 0 , and replace  y  with  \frac{x}{x-1}  to get:  (x)(\frac{x}{x-1})>0 
Simplify:  \frac{x^2}{x-1}>0

Since  x^2  is always greater or equal to 0, it must be the case that the denominator,  x-1 , is positive
In other words, it must be the case that:  x-1>0 
Add  1  to both sides of the inequality to get:  x>1

Answer: D

Cheers, 
Brent

nick1816 · Answer

xy = x + y

xy-y = x

y=\frac{x}{(x-1)}

xy> 0

\frac{x^2}{x-1 }>0

x^2  is always greater than 0; hence x-1>0 or x>1.

D

Kinshook · Answer

Asked: What values of x have a corresponding value of y that satisfies both xy > 0 and xy = x + y ?

Since xy = x+y
xy - y = x
y(x-1) = x
y = x/(x-1)

Since xy > 0
x^2/(x-1)>0
 x 
eq 0 
x-1>0
x>1

IMO D

abhish27 · Answer

uc26 · Answer

Hi  BrentGMATPrepNow ,

Thank you for the solution.
Just wanted to point out a small calculation error in the highlighted part. y should be 2. 
Could you please amend the solution.

Regards,
Udit

BrentGMATPrepNow · Answer

Thanks for the heads up!!
I have edited my solution accordingly.

Kudos for you!!!!

Cheers,
Brent

NitishJain · Answer

xy = x+y
xy - y = x
or y = x/(x-1)

Given:: xy>0
==> x*x/(x-1) > 0
==> x^2 / (x-1) > 0

x^2 will always be +ve

so eqn will be +ve when x-1>0
or x>1

=>  D

AdmissioNinja · Answer

Simplest approach -

Look at option A:

X <= -1  (If X is less than 1 then for the product of X and Y to be greater than zero, Y will have to be negative, which means product of X and Y will not be equal to sum of X and Y, since the product will be positive and sum will be negative)

So A and E are out.

Now look at D, and take X and Y as 2, you have an answer.

Hence D.

unraveled · Answer

BrentGMATPrepNow 
A small question.
Equation xy = x + y with xy > 0 is satisfied by x = y = 2(if only integers are considered). Knowing that helps since you need not calculate algebraically.
Only option D suffices the requirement.

Will this be fine to go ahead with?

BrentGMATPrepNow · Answer

I love that approach - great work!!
The only thing I'll mention is that x = 2 satisfies answer choice D AND answer choice E. 
So you'll still need to test another value of x.

Cheers,
Brent

unraveled · Answer

Hey Brent,
Thanks for a quick response. And sorry if I haven't been clear enough to put my thoughts properly.
I doubt if E would ever be our answer, otherwise why the condition xy > 0. Real Numbers include '0' but if either of x and y are '0' the condition xy > 0 is invalidated.
So, if we can't test '0' then E can never be a contender for an answer.

Hope you are getting my point.

BrentGMATPrepNow · Answer

Good point; since the x = 0 can never satisfy the condition xy > 0, we can rule out answer choice E. 
I was just commenting on the fact that we can't rule out answer choice E simply because x = 2 is one solution.

unraveled · Answer

Yeah..
Many thanks.
I liked your solution - its the best - but just to be sure whether what i did was acceptable or not, I asked.
Also, the query was because I had never eliminated any option - like E in this question - upfront while solving the question and then marked the answer among the four left.

Kudos for such a nice explanation.

davidbeckham · Answer

What does 'corresponding value of x' mean?

gmat2021prep · Answer

(1) xy>0 : numbers could be both +ve or -ve

(2) xy= x+y 
y = (x+y)/x

Since the numbers need to satisfy both, substitute y = (x+y)/x into (1).

x * (x+y)/x > 0 
x+y >0

Soln. needs to be positive = (D).

frankgraves · Answer

Alternative approach, almost without calculation :
Start from xy > 0, we know that x cannot be 0.
So we can eliminate A, B, and E, where x can equal 0.

That leaves us with C or D.
Notice that x may equal 1 in C, but x cannot be 1 in D.
So we should check whether x can be 1.
We know that xy = x + y.
If we substitute x = 1, the equation becomes y = 1 + y and then 0 = 1 (!!!???)
Now we can conclude that x cannot be 1 and we can eliminate C.
D is the remaining possible answer and it is the correct one.

parth2424 · Answer

Hey Brent, Please let me know if I am wrong here. 
My approach is the following:
Corresponding values mean 2 same no.'s with a different unit of measure attached to em. eg 24cm & 24m 
in our case, there are no units of measures 
so we have 2 same no.'s

basically x = y 
xy>0 --> x^2>0 
xy = x+y --> x^2 = 2x 
A VALUE OF X THAT SATISFIES X^2 =2X is 2 HENCE ans is D.

Crytiocanalyst · Answer

xy = x + y

xy-y = x

y=x/(x−1)

xy> 0

x2/x−1>0

x>1

and one brute force checking we can complete by substituing x=y=2 
Therefore IMO D

TusharViv · Answer

What values of x have a corresponding value of y that satisfies both xy > 0 and xy = x + y ?

A. x≤1

B. −1<x≤0

C. 0<x≤1

D. x>1

E. All real numbers

xy > 0 this means either:

1. x>0 AND y>0
2. x<0 AND y<0

xy = x + y

Now the other part of the question. Therefore x+ y > 0 since xy > 0

x + y > 0 can only be true if x or y are positive or both. And since we have the 2 statements above. x<0 AND y<0 can be eliminated.

Therefore eliminate A, B, E

Now between C and D. Get y on one side

xy = x + y

xy - y = x

y(x-1) = x

y = x / (x-1)

x cannot be 1. Eliminate C Hence answer is D
Also if x=1 was not part of C. The denominator would be negative if x was a fraction between 0 and 1, hence giving a negative y value but we know this cannot be true since BOTH x AND y must be positive

What values of x have a corresponding value of y that satisfies both x

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What values of x have a corresponding value of y that satisfies both x

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