Algebra Examples

$\frac{x y - y}{x - 2} - \frac{y}{x + 2}$

Step 1

Factor $y$ out of $x y - y$ .

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Step 1.1

Factor $y$ out of $x y$ .

$\frac{y x - y}{x - 2} - \frac{y}{x + 2}$

Step 1.2

Factor $y$ out of $- y$ .

$\frac{y x + y \cdot - 1}{x - 2} - \frac{y}{x + 2}$

Step 1.3

Factor $y$ out of $y x + y \cdot - 1$ .

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

Step 2

To write $\frac{y (x - 1)}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

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

Step 3

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

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

Step 4

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

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Step 4.1

Multiply $\frac{y (x - 1)}{x - 2}$ by $\frac{x + 2}{x + 2}$ .

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $(x - 2) (x + 2)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Factor $y$ out of $y (x - 1) (x + 2) - y (x - 2)$ .

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Step 6.1.1

Factor $y$ out of $y (x - 1) (x + 2)$ .

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

Step 6.1.2

Factor $y$ out of $- y (x - 2)$ .

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

Step 6.1.3

Factor $y$ out of $y ((x - 1) (x + 2)) + y (- (x - 2))$ .

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

Step 6.2

Expand $(x - 1) (x + 2)$ using the FOIL Method.

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Step 6.2.1

Apply the distributive property.

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

Step 6.2.2

Apply the distributive property.

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

Step 6.2.3

Apply the distributive property.

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

Step 6.3

Simplify and combine like terms.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Multiply $x$ by $x$ .

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

Step 6.3.1.2

Move $2$ to the left of $x$ .

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

Step 6.3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 6.3.1.4

Multiply $- 1$ by $2$ .

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

Step 6.3.2

Subtract $x$ from $2 x$ .

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Multiply $- 1$ by $- 2$ .

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

Step 6.6

Subtract $x$ from $x$ .

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

Step 6.7

Add $x^{2}$ and $0$ .

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

Step 6.8

Add $- 2$ and $2$ .

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

Step 6.9

Add $x^{2}$ and $0$ .

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