Question

X/x-y - x/x+y / xy/x+y

Answer 1

$\frac{\frac{x}{x-y}-\frac{x}{x+y}}{\frac{xy}{x+y}}=\frac{x(x+y)-x(x-y)}{(x-y)(x+y)}*\frac{x+y}{xy}=\frac{x^2+xy-x^2+xy}{(x-y)(x+y)}*\frac{x+y}{xy}=\\ =\frac{2xy}{x-y}*\frac{1}{xy}=\frac{2}{x-y}$

Answer 2

To simplify the expression:

X/(x - y) - x/(x + y) / xy/(x + y)

First, let's simplify each fraction separately:

X/(x - y) can be multiplied by (x + y)/(x + y) to get (X * (x + y))/(x^2 - y^2)

x/(x + y) can be multiplied by (x - y)/(x - y) to get (x * (x - y))/(x^2 - y^2)

xy/(x + y) remains the same.

Now, the expression becomes:

(X * (x + y))/(x^2 - y^2) - (x * (x - y))/(x^2 - y^2) / xy/(x + y)

Next, we can combine the fractions by finding the common denominator, which is (x^2 - y^2):

((X * (x + y)) - (x * (x - y))) / (x^2 - y^2) / xy/(x + y)

Now, let's simplify the numerator:

(X * (x + y)) - (x * (x - y)) = Xx + XY - x^2 + xy

The expression becomes:

(Xx + XY - x^2 + xy) / (x^2 - y^2) / xy/(x + y)

To divide by a fraction, we can multiply by its reciprocal:

(Xx + XY - x^2 + xy) / (x^2 - y^2) * (x + y)/xy

Now, let's simplify further:

(x * (X + Y) - x^2 + xy) / (x^2 - y^2) * (x + y)/xy

We can cancel out common factors:

(x * (X + Y - x + y)) / (x^2 - y^2) * (x + y)/xy

(x * (X + Y - x + y)) / (x^2 - y^2) * (x + y)/(x * y)

Now, we can simplify the expression further by canceling out common factors:

(X + Y - x + y) / (x - y) * (x + y)/y

Finally, we can simplify the expression:

(X + Y - x + y) * (x + y) / (x - y) * y = (X + Y - x + y) * (x + y) * y / (x - y)

So, the simplified expression is (X + Y - x + y) * (x + y) * y / (x - y).