Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $y^{2}$ .

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Step 1.1

Multiply $\frac{\frac{x}{y} - 1}{1 - \frac{x^{2}}{y^{2}}}$ by $\frac{y^{2}}{y^{2}}$ .

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $y$ .

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $y^{2}$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Factor $y$ out of $y x + y^{2} \cdot - 1$ .

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Step 4.1.1

Factor $y$ out of $y x$ .

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

Step 4.1.2

Factor $y$ out of $y^{2} \cdot - 1$ .

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

Step 4.1.3

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $y^{2} \cdot 1$ as ${(y \cdot 1)}^{2}$ .

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

Step 5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y \cdot 1$ and $b = x$ .

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

Step 5.3

Simplify.

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Step 5.3.1

Multiply $y$ by $1$ .

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

Step 5.3.2

Multiply $y$ by $1$ .

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

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

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Step 6.1.1

Factor $- 1$ out of $x$ .

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

Step 6.1.2

Factor $- 1$ out of $- y$ .

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

Step 6.1.3

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

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

Step 6.1.4

Reorder terms.

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

Step 6.1.5

Cancel the common factor.

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

Step 6.1.6

Rewrite the expression.

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

Step 6.2

Simplify the expression.

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Step 6.2.1

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

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

Step 6.2.2

Move the negative in front of the fraction.

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