Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $x y$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Reorder the factors of $y x$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Simplify each term.

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

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

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Step 6.1.2.1

Move $y$ .

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

Step 6.1.2.2

Multiply $y$ by $y$ .

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

Step 6.1.3

Apply the distributive property.

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

Step 6.1.4

Multiply $- - x$ .

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Step 6.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 6.1.4.2

Multiply $x$ by $1$ .

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

Step 6.1.5

Apply the distributive property.

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

Step 6.1.6

Multiply $x$ by $x$ .

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

Step 6.2

Combine the opposite terms in $x y - y^{2} - y x + x^{2} + x^{2} + y^{2}$ .

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Step 6.2.1

Reorder the factors in the terms $x y$ and $- y x$ .

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

Step 6.2.2

Subtract $x y$ from $x y$ .

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

Step 6.2.3

Add $- y^{2}$ and $0$ .

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

Step 6.2.4

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

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

Step 6.2.5

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

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

Step 6.3

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

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

Step 7

Cancel the common factor of $x^{2}$ and $x$ .

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Step 7.1

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

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

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $x$ out of $x y$ .

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

$\frac{2 x}{y}$