Algebra Examples

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

Step 1

To write $\frac{1}{y}$ as a fraction with a common denominator, multiply by $\frac{x - y}{x - y}$ .

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

Step 2

To write $\frac{2}{x - y}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Add $- y$ and $2 y$ .

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

Step 6

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

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

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Step 8.1

Apply the distributive property.

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

Step 8.2

Multiply $x$ by $x$ .

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

Step 8.3

Apply the distributive property.

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

Step 8.4

Subtract $x^{2}$ from $x^{2}$ .

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

Step 8.5

Add $x y$ and $0$ .

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

Step 8.6

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

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Step 8.6.1

Factor $y$ out of $x y$ .

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

Step 8.6.2

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

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

Step 8.6.3

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

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

Step 9

Reduce the expression by cancelling the common factors.

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Step 9.1

Cancel the common factor of $x + y$ .

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Step 9.1.1

Cancel the common factor.

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

Step 9.1.2

Rewrite the expression.

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

Step 9.2

Cancel the common factor of $y (x - y)$ .

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Step 9.2.1

Cancel the common factor.

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

Step 9.2.2

Rewrite the expression.

$1$