Trigonometry Examples

$\frac{6}{y^{2} - x y} - \frac{6}{x^{2} - x y}$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

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

$\frac{6}{y \cdot y - x y} - \frac{6}{x^{2} - x y}$

Step 1.1.2

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

$\frac{6}{y \cdot y + y (- x)} - \frac{6}{x^{2} - x y}$

Step 1.1.3

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

$\frac{6}{y (y - x)} - \frac{6}{x^{2} - x y}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

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

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Step 1.2.1.1

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

$\frac{6}{y (y - x)} - \frac{6}{x \cdot x - x y}$

Step 1.2.1.2

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

$\frac{6}{y (y - x)} - \frac{6}{x \cdot x + x (- 1 y)}$

Step 1.2.1.3

Factor $x$ out of $x \cdot x + x (- 1 y)$ .

$\frac{6}{y (y - x)} - \frac{6}{x (x - 1 y)}$

Step 1.2.2

Rewrite $- 1 y$ as $- y$ .

$\frac{6}{y (y - x)} - \frac{6}{x (x - y)}$

Step 2

Simplify with factoring out.

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Step 2.1

Factor $- 1$ out of $x$ .

$\frac{6}{y (y - x)} - \frac{6}{x (- 1 (- x) - y)}$

Step 2.2

Factor $- 1$ out of $- y$ .

$\frac{6}{y (y - x)} - \frac{6}{x (- 1 (- x) - (y))}$

Step 2.3

Factor $- 1$ out of $- 1 (- x) - (y)$ .

$\frac{6}{y (y - x)} - \frac{6}{x (- 1 (- x + y))}$

Step 2.4

Reorder terms.

$\frac{6}{y (- x + y)} - \frac{6}{x (- 1 (- x + y))}$

Step 3

To write $\frac{6}{y (- x + y)}$ as a fraction with a common denominator, multiply by $\frac{x \cdot - 1}{x \cdot - 1}$ .

$\frac{6}{y (- x + y)} \cdot \frac{x \cdot - 1}{x \cdot - 1} - \frac{6}{x (- 1 (- x + y))}$

Step 4

To write $- \frac{6}{x (- 1 (- x + y))}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{6}{y (- x + y)} \cdot \frac{x \cdot - 1}{x \cdot - 1} - \frac{6}{x (- 1 (- x + y))} \cdot \frac{y}{y}$

Step 5

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

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Step 5.1

Multiply $\frac{6}{y (- x + y)}$ by $\frac{x \cdot - 1}{x \cdot - 1}$ .

$\frac{6 (x \cdot - 1)}{y (- x + y) (x \cdot - 1)} - \frac{6}{x (- 1 (- x + y))} \cdot \frac{y}{y}$

Step 5.2

Multiply $\frac{6}{x (- 1 (- x + y))}$ by $\frac{y}{y}$ .

$\frac{6 (x \cdot - 1)}{y (- x + y) (x \cdot - 1)} - \frac{6 y}{x (- 1 (- x + y)) y}$

Step 5.3

Reorder the factors of $y (- x + y) (x \cdot - 1)$ .

$\frac{6 (x \cdot - 1)}{- x y (- x + y)} - \frac{6 y}{x (- 1 (- x + y)) y}$

Step 5.4

Reorder the factors of $x (- 1 (- x + y)) y$ .

$\frac{6 (x \cdot - 1)}{- x y (- x + y)} - \frac{6 y}{- x y (- x + y)}$

Step 6

Combine the numerators over the common denominator.

$\frac{6 (x \cdot - 1) - 6 y}{- x y (- x + y)}$

Step 7

Simplify the numerator.

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Step 7.1

Factor $6$ out of $6 x \cdot - 1 - 6 y$ .

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Step 7.1.1

Factor $6$ out of $6 x \cdot - 1$ .

$\frac{6 (x \cdot - 1) - 6 y}{- x y (- x + y)}$

Step 7.1.2

Factor $6$ out of $- 6 y$ .

$\frac{6 (x \cdot - 1) + 6 (- y)}{- x y (- x + y)}$

Step 7.1.3

Factor $6$ out of $6 (x \cdot - 1) + 6 (- y)$ .

$\frac{6 (x \cdot - 1 - y)}{- x y (- x + y)}$

Step 7.2

Move $- 1$ to the left of $x$ .

$\frac{6 (- 1 \cdot x - y)}{- x y (- x + y)}$

Step 7.3

Rewrite $- 1 x$ as $- x$ .

$\frac{6 (- x - y)}{- x y (- x + y)}$

Step 8

Simplify terms.

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Step 8.1

Move the negative in front of the fraction.

$- \frac{6 (- x - y)}{x y (- x + y)}$

Step 8.2

Factor $- 1$ out of $- x$ .

$- \frac{6 (- (x) - y)}{x y (- x + y)}$

Step 8.3

Factor $- 1$ out of $- y$ .

$- \frac{6 (- (x) - (y))}{x y (- x + y)}$

Step 8.4

Factor $- 1$ out of $- (x) - (y)$ .

$- \frac{6 (- (x + y))}{x y (- x + y)}$

Step 8.5

Rewrite $- (x + y)$ as $- 1 (x + y)$ .

$- \frac{6 (- 1 (x + y))}{x y (- x + y)}$

Step 8.6

Factor $- 1$ out of $- x$ .

$- \frac{6 (- 1 (x + y))}{x y (- (x) + y)}$

Step 8.7

Factor $- 1$ out of $y$ .

$- \frac{6 (- 1 (x + y))}{x y (- (x) - 1 (- y))}$

Step 8.8

Factor $- 1$ out of $- (x) - 1 (- y)$ .

$- \frac{6 (- 1 (x + y))}{x y (- (x - y))}$

Step 8.9

Rewrite $- (x - y)$ as $- 1 (x - y)$ .

$- \frac{6 (- 1 (x + y))}{x y (- 1 (x - y))}$

Step 8.10

Cancel the common factor.

$- \frac{6 (- 1 (x + y))}{x y (- 1 (x - y))}$

Step 8.11

Rewrite the expression.

$- \frac{6 (x + y)}{x y (x - y)}$