Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Reorder the factors of $y x$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $1$ .

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

Step 5.5

Rewrite $y \cdot y$ as $y^{2}$ .

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

Step 5.6

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

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