Algebra Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Reorder the factors of $y x$ .

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

Simplify the denominator.

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Step 4.2.1

Write $1$ as a fraction with a common denominator.

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

Step 4.2.2

Combine the numerators over the common denominator.

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

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.4

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

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

Step 4.5

Cancel the common factor of $x y$ .

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Step 4.5.1

Cancel the common factor.

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

Step 4.5.2

Rewrite the expression.

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

Step 4.6

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

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

Step 4.7

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

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Step 4.7.1

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

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

Step 4.7.2

Combine.

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

Step 4.8

Apply the distributive property.

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

Step 4.9

Cancel the common factor of $x$ .

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Step 4.9.1

Cancel the common factor.

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

Step 4.9.2

Rewrite the expression.

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

Step 4.10

Factor $x$ out of $x y + x \cdot 1$ .

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Step 4.10.1

Factor $x$ out of $x y$ .

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

Step 4.10.2

Factor $x$ out of $x \cdot 1$ .

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

Step 4.10.3

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $x$ by $1$ .

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

Step 7

Simplify by adding terms.

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Step 7.1

Add $x$ and $x$ .

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

Step 7.2

Simplify the expression.

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Step 7.2.1

Move $1$ .

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

Step 7.2.2

Reorder $2 x$ and $x y$ .

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

Step 8

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

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

Step 9

Simplify terms.

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Step 9.1

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

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

Step 9.2

Combine the numerators over the common denominator.

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

Step 10

Simplify the numerator.

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Step 10.1

Apply the distributive property.

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

Step 10.2

Multiply $- 1$ by $1$ .

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

Step 10.3

Add $- x y$ and $x y$ .

$\frac{0 - 1 + 2 x + 1}{x y + 1}$

Step 10.4

Subtract $1$ from $0$ .

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

Step 10.5

Add $- 1$ and $1$ .

$\frac{2 x + 0}{x y + 1}$

Step 10.6

Add $2 x$ and $0$ .

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