To simplify the expression \(\frac{x^2 - y^2}{2y^2 - 2xy}\), we can start by factoring both the numerator and the denominator.
Step 1: Factor the numerator
The numerator \(x^2 - y^2\) is a difference of squares, which can be factored as follows:
\[ x^2 - y^2 = (x - y)(x + y) \]
Step 2: Factor the denominator
The denominator \(2y^2 - 2xy\) has a common factor of \(2y\):
\[ 2y^2 - 2xy = 2y(y - x) \]
Step 3: Rewrite the expression with the factored forms
Substituting the factored forms back into the expression gives us:
\[ \frac{x^2 - y^2}{2y^2 - 2xy} = \frac{(x - y)(x + y)}{2y(y - x)} \]
Step 4: Simplify the expression
Notice that \(y - x\) can be rewritten as \(-(x - y)\):
\[ \frac{(x - y)(x + y)}{2y(y - x)} = \frac{(x - y)(x + y)}{2y(-1)(x - y)} = -\frac{(x + y)}{2y} \]
Since \((x - y)\) cancels out from the numerator and denominator, we are left with:
\[ -\frac{x + y}{2y} \]
Final Answer:
Thus, the simplest form of the expression is:
\[ -\frac{x + y}{2y} \]
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