Algebra Examples

$\frac{x - y}{x} - \frac{5 y}{x^{2}} \cdot \frac{x^{2} - x y}{5 y}$

Step 1

Simplify each term.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

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

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Step 1.1.1.1

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

$\frac{x - y}{x} - \frac{5 y}{x^{2}} \cdot \frac{x \cdot x - x y}{5 y}$

Step 1.1.1.2

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

$\frac{x - y}{x} - \frac{5 y}{x^{2}} \cdot \frac{x \cdot x + x (- 1 y)}{5 y}$

Step 1.1.1.3

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

$\frac{x - y}{x} - \frac{5 y}{x^{2}} \cdot \frac{x (x - 1 y)}{5 y}$

Step 1.1.2

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

$\frac{x - y}{x} - \frac{5 y}{x^{2}} \cdot \frac{x (x - y)}{5 y}$

Step 1.2

Cancel the common factor of $5 y$ .

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Step 1.2.1

Move the leading negative in $- \frac{5 y}{x^{2}}$ into the numerator.

$\frac{x - y}{x} + \frac{- 5 y}{x^{2}} \cdot \frac{x (x - y)}{5 y}$

Step 1.2.2

Factor $5 y$ out of $- 5 y$ .

$\frac{x - y}{x} + \frac{5 y (- 1)}{x^{2}} \cdot \frac{x (x - y)}{5 y}$

Step 1.2.3

Cancel the common factor.

$\frac{x - y}{x} + \frac{5 y \cdot - 1}{x^{2}} \cdot \frac{x (x - y)}{5 y}$

Step 1.2.4

Rewrite the expression.

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

Step 1.3

Cancel the common factor of $x$ .

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Step 1.3.1

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

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

Step 1.3.2

Cancel the common factor.

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

Step 1.3.3

Rewrite the expression.

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

Step 1.4

Move the negative in front of the fraction.

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

Step 1.5

Apply the distributive property.

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

Step 1.6

Cancel the common factor of $x$ .

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Step 1.6.1

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

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

Step 1.6.2

Cancel the common factor.

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

Step 1.6.3

Rewrite the expression.

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

Step 1.7

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

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Step 1.7.1

Multiply $- 1$ by $- 1$ .

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

Step 1.7.2

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

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

Step 1.7.3

Combine $\frac{1}{x}$ and $y$ .

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

Step 1.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 1.9

Combine $- 1$ and $\frac{x}{x}$ .

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

Step 1.10

Combine the numerators over the common denominator.

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

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Combine the opposite terms in $x - y - x + y$ .

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Step 2.2.1

Subtract $x$ from $x$ .

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

Step 2.2.2

Add $- y$ and $0$ .

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

Step 2.2.3

Add $- y$ and $y$ .

$\frac{0}{x}$

Step 2.3

Divide $0$ by $x$ .

$0$