Algebra Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

Combine $- x$ and $\frac{x - y}{x - y}$ .

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 3

Simplify the numerator.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.2.1

Move $x$ .

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

Step 3.2.2

Multiply $x$ by $x$ .

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

Simplify each term.

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Step 3.4.1

Multiply $- 1$ by $- 1$ .

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

Step 3.4.2

Multiply $x$ by $1$ .

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

Step 3.5

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

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

Step 3.6

Add $x y$ and $0$ .

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

Step 3.7

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

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Step 3.7.1

Factor $y$ out of $x y$ .

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

Step 3.7.2

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

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

Step 3.7.3

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

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

Step 4

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

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

Step 5

Simplify terms.

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Step 5.1

Combine $- y$ and $\frac{x - y}{x - y}$ .

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $- - y$ .

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Step 6.3.1

Multiply $- 1$ by $- 1$ .

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

Step 6.3.2

Multiply $y$ by $1$ .

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

Step 6.4

Add $- x$ and $x$ .

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

Step 6.5

Add $y$ and $0$ .

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

Step 6.6

Add $y$ and $y$ .

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

Step 6.7

Combine exponents.

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Step 6.7.1

Raise $y$ to the power of $1$ .

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

Step 6.7.2

Raise $y$ to the power of $1$ .

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

Step 6.7.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.7.4

Add $1$ and $1$ .

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