Simplify: (x2 - y2) / (x+y) ÷ (2y-2x) / (3xy)

let's simplify this algebraic expression.

Step 1: Rewrite the division as multiplication by the reciprocal of the second fraction. $\frac{x^2 - y^2}{x^2 + xy} \div \frac{2y - 2x}{xy} = \frac{x^2 - y^2}{x^2 + xy} \times \frac{xy}{2y - 2x}$

Step 2: Factorize the numerators and denominators. The numerator $x^2 - y^2$ is a difference of squares, which factors to $(x-y)(x+y)$. The denominator $x^2 + xy$ has a common factor of $x$, which factors to $x(x+y)$. The numerator $2y - 2x$ has a common factor of $2$, which factors to $2(y-x)$. We can also write $y-x$ as $-(x-y)$. The denominator $xy$ is already in its simplest form.

Substitute the factored forms into the expression: $\frac{(x-y)(x+y)}{x(x+y)} \times \frac{xy}{2(-(x-y))}$ $\frac{(x-y)(x+y)}{x(x+y)} \times \frac{xy}{-2(x-y)}$

Step 3: Cancel out common factors from the numerator and denominator. We can cancel $(x+y)$ from the first fraction's numerator and denominator. We can cancel $x$ from the denominator of the first fraction and the numerator of the second fraction. We can cancel $(x-y)$ from the numerator of the first fraction and the denominator of the second fraction.

$\frac{\cancel{(x-y)}\cancel{(x+y)}}{\cancel{x}\cancel{(x+y)}} \times \frac{\cancel{x}y}{-2\cancel{(x-y)}}$

Step 4: Write the simplified expression. $1 \times \frac{y}{-2} = -\frac{y}{2}$

The simplified expression is: -\frac{y{2}} Drop the next question 📸