If 6xy=x^2y+9y, what is the value of xy?

[victoria.babin]

If 6xy=x^2y+9y, what is the value of xy?

1) x= -2
2) x < 0

I understand that the answer to this question is D. My question is about the rephrase given in the explanation, which after factoring, shows that y(x-3)^2=0, so either y=0 or x=3.

The explanation then says that from this we know that xy is either 0 or 3y, so then we can rephrase again to "What is y?" or "Does x=3?"

Both answer choices show that y=0 and are therefore sufficient, but my question is, if one of them did show that x=3, that still wouldn't be sufficient, because then we only have the value of xy=3y, which, without knowing y, is not a real value. Right?

Thanks.

[saurav.raaj]

It gets tricky here.You needn't rephrase the question.

after getting to y * (x-3)^2 = 0, use the conditions 1 and 2. You will arrive at y=0 for both cases.

1. X=-2, means y * (some number)^2=0, hence y=0
2. X<0, means y * (some number)^2=0, hence y=0

for both cases xy=0.

[JonathanSchneider]

True, Saurav, and a nice explanation.


Victoria, your are correct. If we knew x = 3, then we would NOT know the value of y. As a result, the question: "is x = 3?" is definitely of interest, though it is not a complete rephrase of this problem. Note, however, that if we know that x is NOT equal to 3, then y must be zero and our work is done.

[karanrob]

Guys,

What am I doing wrong (conceptually Iam confused):

Given, 6xy=x^2y + 9y
Taking y common on right side:

=>6xy=y(x^2 + 9)----- y cancels on both sides:
x^2 - 6x +9 = 0
x=3

Plugging value of x=3 back in whats given does not give any logical output for y????

[pranabiitkgp]

Karan - you suddenly can not cancel Y from both side . Indeed you seldom cancel any unknown varibale from both side of an euation - only when you are sure that the variable is positive no.

As in this case Y is zero/0 , cancelling means you are deviding the same no. from both side : You can not devide by 0. Hence this technique is itself wrong . Also in inequalities you can not cancel any negative no. from both side which again has implication on the greather than - less than sign .

Thanks,
PM.

[tim]

thanks; let us know if there are any further questions on this one..

[curtis.n.terruptus]

pranabiitkgp. I too canceled out the Y from the equations while simplifying the statement. I don't understand why this is an illegal operation for non-zero values. Would you mind explaining further?

[tim]

It’s not an illegal operation for non-zero values, so if you know FOR SURE that y is not zero you can divide by y. If you are not 100% SURE, you cannot divide by y because you MIGHT be doing something illegal. Better to be on the safe side..