Pre-Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Reorder the factors of $(x - y) (x + y)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $1$ .

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

Step 5.5

Rewrite $(x + y) (x + y)$ as ${(x + y)}^{2}$ .

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

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 $a = x - y$ and $b = x + y$ .

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

Step 5.7

Simplify.

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Step 5.7.1

Add $x$ and $x$ .

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

Step 5.7.2

Add $- y$ and $y$ .

$\frac{(2 x + 0) (x - y - (x + y))}{(x + y) (x - y)}$

Step 5.7.3

Add $2 x$ and $0$ .

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

Step 5.7.4

Apply the distributive property.

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

Step 5.7.5

Subtract $x$ from $x$ .

$\frac{2 x (- y + 0 - y)}{(x + y) (x - y)}$

Step 5.7.6

Add $- y$ and $0$ .

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

Step 5.7.7

Subtract $y$ from $- y$ .

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

Step 5.7.8

Multiply $- 2$ by $2$ .

$\frac{- 4 x y}{(x + y) (x - y)}$

Step 6

Move the negative in front of the fraction.

$- \frac{4 x y}{(x + y) (x - y)}$