Basic Math Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $(a - 2 y) (a + 2 y)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Reorder the factors of $(a - 2 y) (a + 2 y)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $a$ by $a$ .

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

Step 5.2.1.2

Rewrite using the commutative property of multiplication.

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

Step 5.2.1.3

Rewrite using the commutative property of multiplication.

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

Step 5.2.1.4

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

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Step 5.2.1.4.1

Move $y$ .

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

Step 5.2.1.4.2

Multiply $y$ by $y$ .

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

Step 5.2.1.5

Multiply $- 1$ by $2$ .

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

Step 5.2.2

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

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Step 5.2.2.1

Move $y$ .

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

Step 5.2.2.2

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

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

Step 5.3

Multiply $a$ by $1$ .

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

Step 5.4

Apply the distributive property.

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

Step 5.5

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

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Step 5.5.1

Apply the distributive property.

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

Step 5.5.2

Apply the distributive property.

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

Step 5.5.3

Apply the distributive property.

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

Step 5.6

Simplify and combine like terms.

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Step 5.6.1

Simplify each term.

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Step 5.6.1.1

Multiply $a$ by $a$ by adding the exponents.

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Step 5.6.1.1.1

Move $a$ .

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

Step 5.6.1.1.2

Multiply $a$ by $a$ .

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

Step 5.6.1.2

Rewrite using the commutative property of multiplication.

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

Step 5.6.1.3

Multiply $- 1$ by $- 2$ .

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

Step 5.6.1.4

Rewrite using the commutative property of multiplication.

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

Step 5.6.1.5

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

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Step 5.6.1.5.1

Move $y$ .

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

Step 5.6.1.5.2

Multiply $y$ by $y$ .

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

Step 5.6.1.6

Multiply $- 1$ by $- 2$ .

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

Step 5.6.2

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

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Step 5.6.2.1

Move $y$ .

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

Step 5.6.2.2

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

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

Step 5.7

Multiply $a$ by $1$ .

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

Step 5.8

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

$\frac{a y - 2 y^{2} + 0 + a y + 2 y^{2}}{(a + 2 y) (a - 2 y)}$

Step 5.9

Add $a y$ and $0$ .

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

Step 5.10

Add $- 2 y^{2}$ and $2 y^{2}$ .

$\frac{a y + a y + 0}{(a + 2 y) (a - 2 y)}$

Step 5.11

Add $a y + a y$ and $0$ .

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

Step 5.12

Add $a y$ and $a y$ .

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