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Tough and Tricky questions: Algebra.

What is the value of \(\frac{1}{x^2 - y^2} - \frac{1}{x^2 + 2xy + y^2}\)?

(1) \(2y = x^2 - y^2\)

(2) \(x + y = 4\)

Kudos for a correct solution. Official Solution:What is the value of \(\frac{1}{x^2 - y^2} - \frac{1}{x^2 + 2xy + y^2}\)? We must determine the value of the expression \(\frac{1}{x^2 - y^2} - \frac{1}{x^2 + 2xy + y^2}\). When a problem presents an opportunity to factor or find a common denominator, it is usually a good idea to do so.

First, factor the denominator of each fraction. The denominator of the fraction on the left is a difference of squares: \(\frac{1}{x^2 - y^2} = \frac{1}{(x + y)(x - y)}\). The denominator of the fraction on the right is the expanded form of the quadratic expression \((x + y)^{2}\): \(\frac{1}{x^2 + 2xy + y^2} = \frac{1}{(x + y)^2}\).

Thus, the fraction can be rewritten: \(\frac{1}{(x + y)(x - y)} - \frac{1}{(x + y)^2}\). The common denominator of these two fractions is \((x + y)^{2}(x - y)\). Multiply the first fraction by \(\frac{x + y}{x + y}\) and the second fraction by \(\frac{x - y}{x - y}\) and rewrite: \(\frac{x + y}{(x + y)^{2}(x - y)} - \frac{x - y}{(x + y)^{2}(x - y)}\).

Combine the terms by subtracting: \(\frac{x + y - (x - y)}{(x + y)^{2}(x - y)} = \frac{2y}{(x + y)^{2}(x - y)}\).

Statement 1 says that \(2y = x^2 - y^2\), or \(2y = (x + y)(x - y)\). Substitute this into the fraction that we derived above: \(\frac{2y}{(x + y)^{2}(x - y)} = \frac{(x + y)(x - y)}{(x + y)^{2}(x - y)}\). Cancel the factors that appear in both the numerator and the denominator, leaving \(\frac{1}{x + y}\). Without more information about \(x\) or \(y\), we cannot determine the value of this fraction. Statement 1 is NOT sufficient. Eliminate answer choices A and D. The correct answer choice is B, C, or E.

Statement 2 says that \(x + y = 4\). In this case, it will be easier to substitute into the expression \(\frac{1}{(x + y)(x - y)} - \frac{1}{(x + y)^2}\). Doing so gives: \(\frac{1}{4(x - y)} - \frac{1}{(4)^2}\). Without more information about \(x\) and \(y\), however, we cannot determine the value of this expression. Statement 2 is NOT sufficient. Eliminate answer choice B. The correct answer choice is either C or E.

When the statements are taken together, statement 1 allows us to simplify the fraction to \(\frac{1}{x + y}\), and statement 2 tells us that \(x + y = 4\). Substituting, we find: \(\frac{1}{x + y} = \frac{1}{4}\). Together, the statements are sufficient to answer the question.

Answer: C.

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