Question

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

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

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

−2x
Start Fraction negative 2 over x End Fraction

2yx+y

Answer 1

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