[tex]( \frac{x + y}{y} - \frac{x}{x + y} ) \div ( \frac{x + y}{x} - \frac{y}{x + y} ) = \frac{x}{y} [/tex]

Предыдущий
вопрос Следующий
вопрос

To simplify this expression, we can first consider each fraction separately.

Let's look at the numerator of the first fraction:

[tex]\frac{x + y}{y} - \frac{x}{x + y}[/tex]

We can find a common denominator, which is y(x+y).

So, the expression becomes:

[tex]\frac{x(x+y)}{y(x+y)} - \frac{y}{y(x+y)}[/tex]

Simplify this expression:

[tex]\frac{x^2 + xy - y}{y(x+y)}[/tex]

Now let's look at the denominator of the second fraction:

[tex]\frac{x + y}{x} - \frac{y}{x+y}[/tex]

We can find a common denominator, which is x(x+y).

So, the expression becomes:

[tex]\frac{y(x+y)}{x(x+y)} - \frac{x}{x(x+y)}[/tex]

Simplify this expression:

[tex]\frac{y^2 + yx - x}{x(x+y)}[/tex]

Now divide the two fractions:

[tex]\frac{x^2 + xy - y}{y(x+y)} \div \frac{y^2 + yx - x}{x(x+y)}[/tex]

We can rewrite this division as multiplication by the reciprocal:

[tex]\frac{x^2 + xy - y}{y(x+y)} \times \frac{x(x+y)}{y^2 + yx - x}[/tex]

Now, simplify by multiplying the fractions:

[tex]\frac{x(x^2 + xy - y)(x+y)}{y(x+y)(y^2 + yx - x)}[/tex]

Simplify the numerator:

[tex]\frac{x^3 + x^2y - xy + x^2y + xy^2 - y^2}{y(x+y)(y^2 + yx - x)}[/tex]

Combine like terms:

[tex]\frac{x^3 + 2x^2y + xy^2 - y^2}{y(x+y)(y^2 + yx - x)}[/tex]

Now, simplify the expression further if possible.