Precalculus Examples

$\frac{3}{x + y} + \frac{x - 5 y}{x^{2} - y^{2}}$

Step 1

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 $b = y$ .

$\frac{3}{x + y} + \frac{x - 5 y}{(x + y) (x - y)}$

Step 2

To write $\frac{3}{x + y}$ as a fraction with a common denominator, multiply by $\frac{x - y}{x - y}$ .

$\frac{3}{x + y} \cdot \frac{x - y}{x - y} + \frac{x - 5 y}{(x + y) (x - y)}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{3}{x + y}$ by $\frac{x - y}{x - y}$ .

$\frac{3 (x - y)}{(x + y) (x - y)} + \frac{x - 5 y}{(x + y) (x - y)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{3 (x - y) + x - 5 y}{(x + y) (x - y)}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{3 x + 3 (- y) + x - 5 y}{(x + y) (x - y)}$

Step 4.2

Multiply $- 1$ by $3$ .

$\frac{3 x - 3 y + x - 5 y}{(x + y) (x - y)}$

Step 4.3

Add $3 x$ and $x$ .

$\frac{4 x - 3 y - 5 y}{(x + y) (x - y)}$

Step 4.4

Subtract $5 y$ from $- 3 y$ .

$\frac{4 x - 8 y}{(x + y) (x - y)}$

Step 4.5

Factor $4$ out of $4 x - 8 y$ .

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Step 4.5.1

Factor $4$ out of $4 x$ .

$\frac{4 (x) - 8 y}{(x + y) (x - y)}$

Step 4.5.2

Factor $4$ out of $- 8 y$ .

$\frac{4 (x) + 4 (- 2 y)}{(x + y) (x - y)}$

Step 4.5.3

Factor $4$ out of $4 (x) + 4 (- 2 y)$ .

$\frac{4 (x - 2 y)}{(x + y) (x - y)}$