Algebra Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

Cancel the common factor of $x - y$ and ${(x - y)}^{2}$ .

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Step 1.1.2.1

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

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

Step 1.1.2.2

Cancel the common factors.

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Step 1.1.2.2.1

Factor $x - y$ out of ${(x - y)}^{2}$ .

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

Step 1.1.2.2.2

Cancel the common factor.

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

Step 1.1.2.2.3

Rewrite the expression.

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

Step 1.2

Simplify terms.

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Step 1.2.1

Factor $- 1$ out of $y$ .

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

Step 1.2.2

Factor $- 1$ out of $- x$ .

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

Step 1.2.3

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

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

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Move a negative from the denominator of $\frac{2 y}{- 1 (- y + x)}$ to the numerator.

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

Step 1.2.4.2

Reorder terms.

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

Step 1.2.5

Combine the numerators over the common denominator.

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

Step 2

Simplify the numerator.

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Step 2.1

Multiply $- 1$ by $2$ .

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

Step 2.2

Subtract $2 y$ from $y$ .

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

Step 3

Cancel the common factor of $x - y$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

$1$