If x and y are positive and x^2y^2 = 18 – 3xy, then x^2 ?

Question

If x and y are positive and x^2 * y^2 = 18 – 3xy, then x^2 =?

A.  \frac{(18-3y)}{y^3}

B.  \frac{18}{y^2}

C.  \frac{18}{(y^2+3y)}

D.  \frac{9}{y^2}

E.  \frac{36}{y^2}

Bunuel · Accepted Answer

(xy)^2+3(xy)-18=0  --> solving for  xy  -->  xy=-6  (not a valid solution:  xy  must be positive as both unknowns are positive) or  xy=3  --> so  xy=3  -->  x=\frac{3}{y}  -->  x^2=\frac{9}{y^2} .

Answer: D.

Hope it's clear.

mbaiseasy · Answer

x^2y^2 + 3xy = 18 
 xy (xy + 3) = 18

I thought of 2 positive numbers such as  n  and  n + 3  whose product is 18 -->  3  and  6

xy = 3 
 x^2 = 9/y^2

Answer: D

Jp27 · Answer

Hi Bunuel - thanks for your swift response.

Where I can get similar question like these for practice?

I have done the roots and equation set in your signature, are there any other sets that I can use for practice? Sincere thanks!

cheers

Bunuel · Answer

Check our question banks (viewforumtags.php) for more questions. DS algebra questions: search.php?search_id=tag&tag_id=29 PS algebra questions: search.php?search_id=tag&tag_id=50 Hope it helps.

mbaiseasy · Answer

x^2y^2+3xy-18=0 
 (xy)^2+3xy-18=0 
 (xy - 3)(xy + 6)=0 
 xy = 3 & xy = -6  Since x and y are positive, we choose xy = 3.

xy = 3 
 x = \frac{3}{y} 
 x^2 = \frac{9}{y^2}

Answer: D

3111987 · Answer

very elegant&simple...if you spot one variable in two :wink:

like it

noTh1ng · Answer

I don't understand why you can not divide by y^2 here? Pls help

Bunuel · Answer

If you divide x^2 * y^2 = 18 – 3xy by y^2 you get x^2 = 18/y^2 - 3/y = (18-3y)/y^2. We don't have (18-3y)/y^2 among answer choices, thus we need other approach.

noTh1ng · Answer

I sometimes don't get the GMAC..

just because it's not among the answer choices doesnt make it wrong imho

KarishmaB · Answer

If an expression is not in the options, it doesn't make the expression wrong. You might just need to manipulate it further.

Here, dividing by y^2 doesn't work:  x^2 * y^2 = 18 – 3xy 
When you divide by y^2, you get  x^2 = 18/y^2 – 3x/y 
How do you separate the x and y since you need to write x^2 in terms of y only?
You will need to divide by xy and then solve for it.

By the way, GMAC is absolutely reasoning based. If something confuses you, especially in Quant, it means you have missed a point.

EMPOWERgmatRichC · Answer

Hi All,

This question has a "quirky" design element to it - it takes a bit longer than normal to solve AND you might have to TEST VALUES TWICE to get to the correct answer.

You could TEST Y = 3, then X = 1. Then, the answer to the question is also 1 (1² = 1). Plugging Y = 3 into the answer choices gets us 2 answers that match:

Answer C = 18/(3² + 9) = 18/18 = 1 
Answer D = 9/3² = 9/9 = 1

None of the other answers matches, so we're down to 2 choices. At this point, you can either guess one of the 2 or go back and TEST a new value.

If we use Y = 2 in the original equation, then we have 
4X² = 18 - 6X 
4X² +6X - 18 = 0 
2X² +3X - 9 = 0

Factoring this down might seem a little strange, but it DOES lead to a solution… 
(X + 3)(2X - 3) = 0 
X = -3 or +3/2

Since we're told that X and Y are both POSITIVE, X = 3/2

So now we plug Y = 2 into the answers and we're looking for (3/2)² = 9/4

Answer C: 18/10 = 9/5 
Answer D: 9/4

Final Answer:  D

GMAT assassins aren't born, they're made, 
Rich

Shrey9 · Answer

@

IT's a really basic thing I want to ask you, but I must before my exam, I feel...
How do you equate the 2 equations here..
like can you tell me, how is (xy-3) (xy+6) derived ..

I understand that once we open the brackets, we get the above equation, also that 1 entity must be negative and 1 be positive as we have +3 and -18.

But, I ask, how 6 and 3.. is there any other way I am missing ? or is this the right way only..

I welcome any sort of help that will make this crystal clear for me

lucajava · Answer

Call  xy = t  and solve the equation. You'll get:

t^2 +3t -18 = 0 
 (t + 6) (t - 3) = 0 .  6 - 3 = 3 ;  6 * -3 = -18

xy = -6 
 xy = 3

Since you consider only positive values,  x = \frac{3}{y} , so  x^2 = \frac{9}{y^2} .

Hope it's clear.

aaditya005 · Answer

"Plug in the value" is a good approach for this one. Take x = 3 and y = 1. Plug in the values and we'll get 9 as the answer. Moreover, from the options only one option D satisfies that.
On the more regular approach, I really like the method of taking xy as one variable T and then soving the quadratic equation thus formed.

P.S. :- This is my first post. Please give kudos if this helps you and help me grow here. Thanks in advance.

bebs · Answer

Let's assume y = 1 ==>  x^{2}  + 3x - 18 = 0 ==> x = -6 or 3 ==> x = 3 since we are told x is positive ==>  x^{2}  =  3^{2}  = 9 ==> test the answers with y = 1, the option that equals 9 is the answer ==> answer is   D

Kritisood · Answer

for ease of calculation let xy = a (a positive integer)
a^2=18-3a
a^2+3a-18=0
(a+6)(a-3)
a=3
a=-6 (not possible as xy is a +ve integer

putting in the equation in the stem
x^2=18-9/y^2
x^2=9/y^2

bumpbot · Answer

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If x and y are positive and x^2y^2 = 18 – 3xy, then x^2 ?

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If x and y are positive and x^2y^2 = 18 – 3xy, then x^2 ?

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