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Advanced Math / Equivalent expressions Difficulty: Hard

Which expression is equivalent to $\frac{y + 12}{x - 8} + \frac{y (x - 8)}{x^{2} y - 8 x y}$ ?

$\frac{x y + y + 4}{x^{3} y - 16 x^{2} y + 64 x y}$

$\frac{x y + 9 y + 12}{x^{2} y - 8 x y + x - 8}$

$\frac{x y^{2} + 13 x y - 8 y}{x^{2} y - 8 x y}$

$\frac{x y^{2} + 13 x y - 8 y}{x^{3} y - 16 x^{2} y + 64 x y}$

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Explanation

Choice C is correct. Factoring the denominator in the second term of the given expression gives $\frac{y + 12}{x - 8} + \frac{y (x - 8)}{x y (x - 8)}$ . This expression can be rewritten with common denominators by multiplying the first term by $\frac{x y}{x y}$ , giving $\frac{x y (y + 12)}{x y (x - 8)} + \frac{y (x - 8)}{x y (x - 8)}$ . Adding these two terms yields $\frac{x y (y + 12) + y (x - 8)}{x y (x - 8)}$ . Using the distributive property to rewrite this expression gives $\frac{x y^{2} + 12 x y + x y - 8 y}{x^{2} y - 8 x y}$ . Combining the like terms in the numerator of this expression gives $\frac{x y^{2} + 13 x y - 8 y}{x^{2} y - 8 x y}$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.