Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.2

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 $b = y$ .

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $a$ by $a$ .

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

Step 4.2.1.2

Rewrite using the commutative property of multiplication.

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

Step 4.2.1.3

Rewrite using the commutative property of multiplication.

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

Step 4.2.1.4

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

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Step 4.2.1.4.1

Move $y$ .

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

Step 4.2.1.4.2

Multiply $y$ by $y$ .

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

Step 4.2.1.5

Multiply $- 2$ by $- 1$ .

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

Step 4.2.2

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

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Step 4.2.2.1

Move $y$ .

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

Step 4.2.2.2

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

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

Step 4.3

Apply the distributive property.

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

Step 4.4

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

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Step 4.4.1

Move $y$ .

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

Step 4.4.2

Multiply $y$ by $y$ .

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

Step 4.5

Rewrite using the commutative property of multiplication.

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

Step 4.6

Multiply $- 1$ by $- 5$ .

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

Step 4.7

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

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Step 4.7.1

Move $y$ .

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

Step 4.7.2

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

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

Step 4.8

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

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

Step 4.9

Factor using the perfect square rule.

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Step 4.9.1

Rearrange terms.

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

Step 4.9.2

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

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

Step 4.9.3

Rewrite the polynomial.

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

Step 4.9.4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Cancel the common factors.

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Rewrite the expression.

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