Finite Math Examples

$\frac{5}{x + y} - \frac{5}{x - y}$

Step 1

To write $\frac{5}{x + y}$ as a fraction with a common denominator, multiply by $\frac{x - y}{x - y}$ .

$\frac{5}{x + y} \cdot \frac{x - y}{x - y} - \frac{5}{x - y}$

Step 2

To write $- \frac{5}{x - y}$ as a fraction with a common denominator, multiply by $\frac{x + y}{x + y}$ .

$\frac{5}{x + y} \cdot \frac{x - y}{x - y} - \frac{5}{x - y} \cdot \frac{x + y}{x + y}$

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 $\frac{5}{x + y}$ by $\frac{x - y}{x - y}$ .

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

Step 3.2

Multiply $\frac{5}{x - y}$ by $\frac{x + y}{x + y}$ .

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Factor $5$ out of $5 (x - y) - 5 (x + y)$ .

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Step 5.1.1

Factor $5$ out of $- 5 (x + y)$ .

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

Step 5.1.2

Factor $5$ out of $5 (x - y) + 5 (- (x + y))$ .

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Subtract $x$ from $x$ .

$\frac{5 (- y + 0 - y)}{(x + y) (x - y)}$

Step 5.4

Add $- y$ and $0$ .

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

Step 5.5

Subtract $y$ from $- y$ .

$\frac{5 \cdot - 2 y}{(x + y) (x - y)}$

Step 5.6

Multiply $5$ by $- 2$ .

$\frac{- 10 y}{(x + y) (x - y)}$

Step 6

Move the negative in front of the fraction.

$- \frac{10 y}{(x + y) (x - y)}$