Solve the equation (x + y) / (x - y) - (x - y) / (x + y) = 2xy / (x^2 - y^2)

Steps to Solve

1. Combine the fractions on the left side To combine the fractions ( \frac{x + y}{x - y} - \frac{x - y}{x + y} ), find a common denominator, which is ( (x - y)(x + y) ): [ \frac{(x + y)^2 - (x - y)^2}{(x - y)(x + y)} ]

2. Expand the numerators Now expand the numerators:

• ( (x + y)^2 = x^2 + 2xy + y^2 )
• ( (x - y)^2 = x^2 - 2xy + y^2 )

Substituting back in gives: [ \frac{(x^2 + 2xy + y^2) - (x^2 - 2xy + y^2)}{(x - y)(x + y)} ]

3. Simplify the numerator Simplifying the numerator: [ x^2 + 2xy + y^2 - x^2 + 2xy - y^2 = 4xy ] Thus, we have: [ \frac{4xy}{(x - y)(x + y)} ]

4. Set the equation with the right side Now our equation reads: [ \frac{4xy}{(x - y)(x + y)} = \frac{2xy}{x^2 - y^2} ]

5. Recognize that ( x^2 - y^2 = (x - y)(x + y) ) Replace ( x^2 - y^2 ) in the right side: [ \frac{4xy}{(x - y)(x + y)} = \frac{2xy}{(x - y)(x + y)} ]

6. Cross multiply Cross-multiply to eliminate the fractions: [ 4xy \cdot (x - y)(x + y) = 2xy \cdot (x - y)(x + y) ]

7. Simplify the equation Since ( (x - y)(x + y) ) is common on both sides and can be divided out (assuming ( x \neq y )): [ 4xy = 2xy ]

8. Divide both sides by ( 2xy ) (assuming ( xy \neq 0 )) This simplifies to: [ 2 = 1 ] This means there's no solution under the assumption that both sides are equal.