Algebra Examples

$\frac{\frac{y}{x} - \frac{x}{y}}{\frac{1}{y} - \frac{1}{x}}$

Step 1

Multiply the numerator and denominator of the fraction by $x y$ .

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Step 1.1

Multiply $\frac{\frac{y}{x} - \frac{x}{y}}{\frac{1}{y} - \frac{1}{x}}$ by $\frac{x y}{x y}$ .

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Factor $x$ out of $x y$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

Cancel the common factor of $y$ .

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Step 3.6.1

Move the leading negative in $- \frac{x}{y}$ into the numerator.

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

Step 3.6.2

Factor $y$ out of $x y$ .

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

Step 3.6.3

Cancel the common factor.

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

Step 3.6.4

Rewrite the expression.

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Add $1$ and $1$ .

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

Step 3.11

Cancel the common factor of $y$ .

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Step 3.11.1

Factor $y$ out of $x y$ .

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

Step 3.11.2

Cancel the common factor.

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

Step 3.11.3

Rewrite the expression.

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

Step 3.12

Cancel the common factor of $x$ .

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Step 3.12.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

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

Step 3.12.2

Factor $x$ out of $x y$ .

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

Step 3.12.3

Cancel the common factor.

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

Step 3.12.4

Rewrite the expression.

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

Step 4

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

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

Step 5

Simplify the denominator.

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Step 5.1

Move $- 1$ to the left of $y$ .

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

Step 5.2

Rewrite $- 1 y$ as $- y$ .

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

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $y - x$ and $x - y$ .

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Step 6.1.1

Factor $- 1$ out of $y$ .

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

Step 6.1.2

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

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

Step 6.1.3

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

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

Step 6.1.4

Reorder terms.

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

Step 6.1.5

Cancel the common factor.

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

Step 6.1.6

Divide $(y + x) \cdot (- 1)$ by $1$ .

$(y + x) \cdot (- 1)$

Step 6.2

Simplify by multiplying through.

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Step 6.2.1

Apply the distributive property.

$y \cdot - 1 + x \cdot - 1$

Step 6.2.2

Reorder.

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Step 6.2.2.1

Move $- 1$ to the left of $y$ .

$- 1 \cdot y + x \cdot - 1$

Step 6.2.2.2

Move $- 1$ to the left of $x$ .

$- 1 \cdot y - 1 \cdot x$

Step 6.3

Simplify each term.

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Step 6.3.1

Rewrite $- 1 y$ as $- y$ .

$- y - 1 \cdot x$

Step 6.3.2

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

$- y - x$