To simplify the expression \( \frac{2y^2 - 2xy}{x^2 - y^2} \), we can factor both the numerator and the denominator.
Step 1: Factor the numerator
The numerator \( 2y^2 - 2xy \) can be factored as:
\[ 2y(y - x) \]
Step 2: Factor the denominator
The denominator \( x^2 - y^2 \) can be factored using the difference of squares:
\[ x^2 - y^2 = (x - y)(x + y) \]
Step 3: Rewrite the expression
Now, we rewrite the expression with the factored numerator and denominator:
\[ \frac{2y(y - x)}{(x - y)(x + y)} \]
Step 4: Simplify the expression
Notice that \( y - x \) can be rewritten as \( -(x - y) \):
\[ \frac{2y(y - x)}{(x - y)(x + y)} = \frac{2y(-1)(x - y)}{(x - y)(x + y)} \]
We can now cancel \( (x - y) \) from the numerator and denominator (assuming \( x \neq y \)):
\[ \frac{-2y}{x + y} \]
So, the simplest form of the expression is:
\[ \frac{-2y}{x + y} \]
Final Answer:
Start Fraction negative 2 y over x plus y End Fraction
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