Finite Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Reorder the factors of $y x$ .

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

Rewrite $\frac{x \cdot x - y \cdot y}{x y}$ in a factored form.

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Step 1.5.1

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

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

Step 1.5.2

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

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

Step 1.5.3

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

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

Step 1.5.4

Add $1$ and $1$ .

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

Step 1.5.5

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

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

Step 1.5.6

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 $b = y$ .

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

Step 1.6

Combine exponents.

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Step 1.6.1

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

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

Step 1.6.2

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

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

Step 1.7

Cancel the common factor of $x$ .

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Step 1.7.1

Cancel the common factor.

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

Step 1.7.2

Rewrite the expression.

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

Step 1.8

Cancel the common factor of $y$ .

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Step 1.8.1

Cancel the common factor.

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

Step 1.8.2

Divide $(x + y) (x - y)$ by $1$ .

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

Step 1.9

Expand $(x + y) (x - y)$ using the FOIL Method.

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Step 1.9.1

Apply the distributive property.

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

Step 1.9.2

Apply the distributive property.

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

Step 1.9.3

Apply the distributive property.

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

Step 1.10

Combine the opposite terms in $x \cdot x + x (- y) + y x + y (- y)$ .

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Step 1.10.1

Reorder the factors in the terms $x (- y)$ and $y x$ .

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

Step 1.10.2

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

$\frac{x \cdot x + 0 + y (- y)}{x + y}$

Step 1.10.3

Add $x \cdot x$ and $0$ .

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

Step 1.11

Simplify each term.

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Step 1.11.1

Multiply $x$ by $x$ .

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

Step 1.11.2

Rewrite using the commutative property of multiplication.

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

Step 1.11.3

Multiply $y$ by $y$ by adding the exponents.

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Step 1.11.3.1

Move $y$ .

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

Step 1.11.3.2

Multiply $y$ by $y$ .

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

Step 1.12

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 $b = y$ .

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

Step 2

Cancel the common factor of $x + y$ .

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Step 2.1

Cancel the common factor.

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

Step 2.2

Divide $x - y$ by $1$ .

$x - y$