Precalculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

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 $x y$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $y$ .

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Step 3.2.1

Factor $y$ out of $x y$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $y$ .

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Step 3.3.1

Factor $y$ out of $x y$ .

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $x$ .

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Step 3.4.1

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

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

Step 3.4.2

Factor $x$ out of $x y$ .

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

Step 3.4.3

Cancel the common factor.

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

Step 3.4.4

Rewrite the expression.

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Simplify the denominator.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

Simplify the denominator.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Add $1$ and $1$ .

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

Step 7

Cancel the common factors.

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Step 7.1

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

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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