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

[Vincen]

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

$$ (1) y-x=3$$

$$ (2) x^3<0$$

The OA is B.

I would appreciate an explanation here. Please.

[Matt@VeritasPrep]

6xy = x²y + 9y

y * 6x = y * (x² + 9)

If y = 0, this statement will be true, so y = 0 is one solution. To find the other, we divide by y:

6x = x² + 9

0 = x² - 6x + 9

0 = (x - 3)²

and x = 3 is our OTHER solution. With that, we know that either y = 0 or x = 3.

S2 tells us that x ≠ 3, so we MUST have y = 0, and x * y = x * 0 = 0.