To simplify the expression \( \frac{2y^2 - 2xy}{x^2 - y^2} \), we can start by factoring both the numerator and the denominator.
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Factor the numerator: \[ 2y^2 - 2xy = 2(y^2 - xy) = 2y(y - x) \]
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Factor the denominator: The denominator \( x^2 - y^2 \) is a difference of squares, which factors as: \[ x^2 - y^2 = (x - y)(x + y) \]
Now we can rewrite the expression: \[ \frac{2y(y - x)}{(x - y)(x + y)} \]
Notice that \( y - x = -(x - y) \), so we can rewrite \( y - x \) and simplify the fraction: \[ \frac{2y(-1)(x - y)}{(x - y)(x + y)} = \frac{-2y(x - y)}{(x - y)(x + y)} \]
Now we can cancel \( (x - y) \) from the numerator and the denominator (as long as \( x \neq y \)): \[ \frac{-2y}{x + y} \]
Thus, the simplest form of the expression is: \[ -\frac{2y}{x + y} \]
Looking at the available responses, the correct answer is: \[ \text{Start Fraction negative 2 y over x plus y End Fraction} \]
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