Question

Express in simplest form: 2y2−2xyx2−y2
.(1 point)
Responses

−2x
Start Fraction negative 2 over x End Fraction

−2yx+y
Start Fraction negative 2 y over x plus y End Fraction

2yx+y
Start Fraction 2 y over x plus y End Fraction

2−2yx
Start Fraction 2 minus 2 y over x End Fraction

Answer 1

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 \(-\frac{2y}{x+y}\).