Pre-Algebra Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Factor $x$ out of $x y$ .

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

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 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Multiply $x$ by $x$ .

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

Step 7.3

Apply the distributive property.

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

Step 7.4

Rewrite using the commutative property of multiplication.

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

Step 7.5

Simplify each term.

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Step 7.5.1

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

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Step 7.5.1.1

Move $y$ .

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

Step 7.5.1.2

Multiply $y$ by $y$ .

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

Step 7.5.2

Multiply $- 1$ by $- 1$ .

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

Step 7.5.3

Multiply $y^{2}$ by $1$ .

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

Step 7.6

Subtract $y x$ from $x y$ .

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Step 7.6.1

Move $y$ .

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

Step 7.6.2

Subtract $x y$ from $x y$ .

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

Step 7.7

Add $x^{2}$ and $0$ .

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

Step 8

Simplify terms.

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Step 8.1

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

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Step 8.1.1

Cancel the common factor.

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

Step 8.1.2

Rewrite the expression.

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

Step 8.2

Cancel the common factor of $x + y$ .

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Step 8.2.1

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

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

Step 8.2.2

Cancel the common factor.

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

Step 8.2.3

Rewrite the expression.

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

Step 8.3

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

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