Simplify (a-2y)/(a+y)-(y^2-5ay)/(a^2-y^2)

$\frac{(a - 2 y)}{a + y} - \frac{(y^{2} - 5 a y)}{a^{2} - y^{2}}$

Simplify each term.

Factor $y$ out of $y^{2} - 5 a y$ .

Factor $y$ out of $y^{2}$ .

$\frac{a - 2 y}{a + y} - \frac{y \cdot y - 5 a y}{a^{2} - y^{2}}$

Factor $y$ out of $- 5 a y$ .

$\frac{a - 2 y}{a + y} - \frac{y \cdot y + y (- 5 a)}{a^{2} - y^{2}}$

Factor $y$ out of $y \cdot y + y (- 5 a)$ .

$\frac{a - 2 y}{a + y} - \frac{y (y - 5 a)}{a^{2} - y^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = y$ .

$\frac{a - 2 y}{a + y} - \frac{y (y - 5 a)}{(a + y) (a - y)}$

To write $\frac{a - 2 y}{a + y}$ as a fraction with a common denominator, multiply by $\frac{a - y}{a - y}$ .

$\frac{a - 2 y}{a + y} \cdot \frac{a - y}{a - y} - \frac{y (y - 5 a)}{(a + y) (a - y)}$

Simplify terms.

Multiply $\frac{a - 2 y}{a + y}$ by $\frac{a - y}{a - y}$ .

$\frac{(a - 2 y) (a - y)}{(a + y) (a - y)} - \frac{y (y - 5 a)}{(a + y) (a - y)}$

Combine the numerators over the common denominator.

$\frac{(a - 2 y) (a - y) - y (y - 5 a)}{(a + y) (a - y)}$

Simplify the numerator.

Expand $(a - 2 y) (a - y)$ using the FOIL Method.

Apply the distributive property.

$\frac{a (a - y) - 2 y (a - y) - y (y - 5 a)}{(a + y) (a - y)}$

Apply the distributive property.

$\frac{a \cdot a + a (- y) - 2 y (a - y) - y (y - 5 a)}{(a + y) (a - y)}$

Apply the distributive property.

$\frac{a \cdot a + a (- y) - 2 y a - 2 y (- y) - y (y - 5 a)}{(a + y) (a - y)}$

Simplify and combine like terms.

Simplify each term.

Multiply $a$ by $a$ .

$\frac{a^{2} + a (- y) - 2 y a - 2 y (- y) - y (y - 5 a)}{(a + y) (a - y)}$

Rewrite using the commutative property of multiplication.

$\frac{a^{2} - a y - 2 y a - 2 y (- y) - y (y - 5 a)}{(a + y) (a - y)}$

Rewrite using the commutative property of multiplication.

$\frac{a^{2} - a y - 2 y a - 2 \cdot - 1 y \cdot y - y (y - 5 a)}{(a + y) (a - y)}$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$\frac{a^{2} - a y - 2 y a - 2 \cdot - 1 (y \cdot y) - y (y - 5 a)}{(a + y) (a - y)}$

Multiply $y$ by $y$ .

$\frac{a^{2} - a y - 2 y a - 2 \cdot - 1 y^{2} - y (y - 5 a)}{(a + y) (a - y)}$

Multiply $- 2$ by $- 1$ .

$\frac{a^{2} - a y - 2 y a + 2 y^{2} - y (y - 5 a)}{(a + y) (a - y)}$

Subtract $2 y a$ from $- a y$ .

Move $y$ .

$\frac{a^{2} - a y - 2 a y + 2 y^{2} - y (y - 5 a)}{(a + y) (a - y)}$

Subtract $2 a y$ from $- a y$ .

$\frac{a^{2} - 3 a y + 2 y^{2} - y (y - 5 a)}{(a + y) (a - y)}$

Apply the distributive property.

$\frac{a^{2} - 3 a y + 2 y^{2} - y \cdot y - y (- 5 a)}{(a + y) (a - y)}$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$\frac{a^{2} - 3 a y + 2 y^{2} - (y \cdot y) - y (- 5 a)}{(a + y) (a - y)}$

Multiply $y$ by $y$ .

$\frac{a^{2} - 3 a y + 2 y^{2} - y^{2} - y (- 5 a)}{(a + y) (a - y)}$

Rewrite using the commutative property of multiplication.

$\frac{a^{2} - 3 a y + 2 y^{2} - y^{2} - 1 \cdot - 5 y a}{(a + y) (a - y)}$

Multiply $- 1$ by $- 5$ .

$\frac{a^{2} - 3 a y + 2 y^{2} - y^{2} + 5 y a}{(a + y) (a - y)}$

Add $- 3 a y$ and $5 y a$ .

Move $y$ .

$\frac{a^{2} + 2 y^{2} - y^{2} - 3 a y + 5 a y}{(a + y) (a - y)}$

Add $- 3 a y$ and $5 a y$ .

$\frac{a^{2} + 2 y^{2} - y^{2} + 2 a y}{(a + y) (a - y)}$

Subtract $y^{2}$ from $2 y^{2}$ .

$\frac{a^{2} + y^{2} + 2 a y}{(a + y) (a - y)}$

Factor using the perfect square rule.

Rearrange terms.

$\frac{a^{2} + 2 a y + y^{2}}{(a + y) (a - y)}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a y = 2 \cdot a \cdot y$

Rewrite the polynomial.

$\frac{a^{2} + 2 \cdot a \cdot y + y^{2}}{(a + y) (a - y)}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = a$ and $b = y$ .

$\frac{{(a + y)}^{2}}{(a + y) (a - y)}$

Cancel the common factor of ${(a + y)}^{2}$ and $a + y$ .

Factor $a + y$ out of ${(a + y)}^{2}$ .

$\frac{(a + y) (a + y)}{(a + y) (a - y)}$

Cancel the common factors.

Cancel the common factor.

$\frac{(a + y) (a + y)}{(a + y) (a - y)}$

Rewrite the expression.

$\frac{a + y}{a - y}$

Simplify (a-2y)/(a+y)-(y^2-5ay)/(a^2-y^2)

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