Algebra Examples

$\frac{\frac{2}{x - y} + \frac{1}{x + y}}{\frac{1}{x - y}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

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

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

Step 3

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

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

Step 4

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 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Add $2 x$ and $x$ .

$\frac{3 x + 2 y - y}{(x + y) (x - y)} (x - y)$

Step 6.3

Subtract $y$ from $2 y$ .

$\frac{3 x + y}{(x + y) (x - y)} (x - y)$

Step 7

Cancel the common factor of $x - y$ .

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Step 7.1

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

$\frac{3 x + y}{(x - y) (x + y)} (x - y)$

Step 7.2

Cancel the common factor.

$\frac{3 x + y}{(x - y) (x + y)} (x - y)$

Step 7.3

Rewrite the expression.

$\frac{3 x + y}{x + y}$