To simplify the expression \(\frac{2y^2 - 2xy}{x^2 - y^2}\), we start by factoring both the numerator and the denominator.
Step 1: Factor the numerator
The numerator is \(2y^2 - 2xy\): \[ 2y^2 - 2xy = 2y(y - x) \]
Step 2: Factor the denominator
The denominator is \(x^2 - y^2\), which is a difference of squares: \[ x^2 - y^2 = (x - y)(x + y) \]
Step 3: Rewrite the expression
Now we can rewrite the expression using the factored forms: \[ \frac{2y^2 - 2xy}{x^2 - y^2} = \frac{2y(y - x)}{(x - y)(x + y)} \] Noticing that \(y - x = -(x - y)\), we can substitute this into the fraction: \[ = \frac{2y(y - x)}{-(x - y)(x + y)} = \frac{-2y(y - x)}{(x - y)(x + y)} = -\frac{2y}{(x + y)} \] since the \(y - x\) cancels out with \(-(x - y)\).
Conclusion
Thus, the simplified form is: \[ -\frac{2y}{x + y} \]
Looking at the response options, this corresponds to: \[ \text{−2yx+y } \text{Start Fraction negative 2 y over x plus y End Fraction} \]
So the correct answer is: \(-\frac{2y}{x + y}\) or \(-\frac{2y}{x+y}\).
3 answers