Algebra Examples

$\frac{2}{x - y} + \frac{3}{x + y} - \frac{5}{y - x} - \frac{7 x - 9 y}{y^{2} - x^{2}}$

Step 1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = x$ .

$\frac{2}{x - y} + \frac{3}{x + y} - \frac{5}{y - x} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 2

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

$\frac{2}{x - y} \cdot \frac{x + y}{x + y} + \frac{3}{x + y} - \frac{5}{y - x} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 3

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

$\frac{2}{x - y} \cdot \frac{x + y}{x + y} + \frac{3}{x + y} \cdot \frac{x - y}{x - y} - \frac{5}{y - x} - \frac{7 x - 9 y}{(y + x) (y - x)}$

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

$\frac{2 (x + y)}{(x - y) (x + y)} + \frac{3}{x + y} \cdot \frac{x - y}{x - y} - \frac{5}{y - x} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 4.2

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

$\frac{2 (x + y)}{(x - y) (x + y)} + \frac{3 (x - y)}{(x + y) (x - y)} - \frac{5}{y - x} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 4.3

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

$\frac{2 (x + y)}{(x + y) (x - y)} + \frac{3 (x - y)}{(x + y) (x - y)} - \frac{5}{y - x} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 5

Combine the numerators over the common denominator.

$\frac{2 (x + y) + 3 (x - y)}{(x + y) (x - y)} - \frac{5}{y - x} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{2 x + 2 y + 3 (x - y)}{(x + y) (x - y)} - \frac{5}{y - x} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 6.2

Apply the distributive property.

$\frac{2 x + 2 y + 3 x + 3 (- y)}{(x + y) (x - y)} - \frac{5}{y - x} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 6.3

Multiply $- 1$ by $3$ .

$\frac{2 x + 2 y + 3 x - 3 y}{(x + y) (x - y)} - \frac{5}{y - x} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 6.4

Add $2 x$ and $3 x$ .

$\frac{5 x + 2 y - 3 y}{(x + y) (x - y)} - \frac{5}{y - x} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 6.5

Subtract $3 y$ from $2 y$ .

$\frac{5 x - y}{(x + y) (x - y)} - \frac{5}{y - x} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 7

Simplify with factoring out.

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Step 7.1

Factor $- 1$ out of $y$ .

$\frac{5 x - y}{(x + y) (x - y)} - \frac{5}{- 1 (- y) - x} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 7.2

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

$\frac{5 x - y}{(x + y) (x - y)} - \frac{5}{- 1 (- y) - (x)} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 7.3

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

$\frac{5 x - y}{(x + y) (x - y)} - \frac{5}{- 1 (- y + x)} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 7.4

Reorder terms.

$\frac{5 x - y}{(x + y) (x - y)} - \frac{5}{- 1 (x - y)} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 8

To write $\frac{5 x - y}{(x + y) (x - y)}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$\frac{5 x - y}{(x + y) (x - y)} \cdot \frac{- 1}{- 1} - \frac{5}{- 1 (x - y)} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 9

To write $- \frac{5}{- 1 (x - y)}$ as a fraction with a common denominator, multiply by $\frac{x + y}{x + y}$ .

$\frac{5 x - y}{(x + y) (x - y)} \cdot \frac{- 1}{- 1} - \frac{5}{- 1 (x - y)} \cdot \frac{x + y}{x + y} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 10

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

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Step 10.1

Multiply $\frac{5 x - y}{(x + y) (x - y)}$ by $\frac{- 1}{- 1}$ .

$\frac{(5 x - y) \cdot - 1}{(x + y) (x - y) \cdot - 1} - \frac{5}{- 1 (x - y)} \cdot \frac{x + y}{x + y} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 10.2

Multiply $\frac{5}{- 1 (x - y)}$ by $\frac{x + y}{x + y}$ .

$\frac{(5 x - y) \cdot - 1}{(x + y) (x - y) \cdot - 1} - \frac{5 (x + y)}{- 1 (x - y) (x + y)} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 10.3

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

$\frac{(5 x - y) \cdot - 1}{- (x + y) (x - y)} - \frac{5 (x + y)}{- 1 (x - y) (x + y)} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 10.4

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

$\frac{(5 x - y) \cdot - 1}{- (x + y) (x - y)} - \frac{5 (x + y)}{- (x + y) (x - y)} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 11

Combine the numerators over the common denominator.

$\frac{(5 x - y) \cdot - 1 - 5 (x + y)}{- (x + y) (x - y)} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 12

Simplify each term.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

Apply the distributive property.

$\frac{5 x \cdot - 1 - y \cdot - 1 - 5 (x + y)}{- (x + y) (x - y)} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 12.1.2

Multiply $- 1$ by $5$ .

$\frac{- 5 x - y \cdot - 1 - 5 (x + y)}{- (x + y) (x - y)} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 12.1.3

Multiply $- y \cdot - 1$ .

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Step 12.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{- 5 x + 1 y - 5 (x + y)}{- (x + y) (x - y)} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 12.1.3.2

Multiply $y$ by $1$ .

$\frac{- 5 x + y - 5 (x + y)}{- (x + y) (x - y)} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 12.1.4

Apply the distributive property.

$\frac{- 5 x + y - 5 x - 5 y}{- (x + y) (x - y)} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 12.1.5

Subtract $5 x$ from $- 5 x$ .

$\frac{- 10 x + y - 5 y}{- (x + y) (x - y)} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 12.1.6

Subtract $5 y$ from $y$ .

$\frac{- 10 x - 4 y}{- (x + y) (x - y)} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 12.1.7

Factor $2$ out of $- 10 x - 4 y$ .

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Step 12.1.7.1

Factor $2$ out of $- 10 x$ .

$\frac{2 (- 5 x) - 4 y}{- (x + y) (x - y)} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 12.1.7.2

Factor $2$ out of $- 4 y$ .

$\frac{2 (- 5 x) + 2 (- 2 y)}{- (x + y) (x - y)} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 12.1.7.3

Factor $2$ out of $2 (- 5 x) + 2 (- 2 y)$ .

$\frac{2 (- 5 x - 2 y)}{- (x + y) (x - y)} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 12.2

Move the negative in front of the fraction.

$- \frac{2 (- 5 x - 2 y)}{(x + y) (x - y)} - \frac{7 x - 9 y}{(y + x) (y - x)}$

Step 13

Simplify with factoring out.

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Step 13.1

Reorder terms.

$- \frac{2 (- 5 x - 2 y)}{(x + y) (x - y)} - \frac{7 x - 9 y}{(x + y) (y - x)}$

Step 13.2

Factor $- 1$ out of $y$ .

$- \frac{2 (- 5 x - 2 y)}{(x + y) (x - y)} - \frac{7 x - 9 y}{(x + y) (- 1 (- y) - x)}$

Step 13.3

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

$- \frac{2 (- 5 x - 2 y)}{(x + y) (x - y)} - \frac{7 x - 9 y}{(x + y) (- 1 (- y) - (x))}$

Step 13.4

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

$- \frac{2 (- 5 x - 2 y)}{(x + y) (x - y)} - \frac{7 x - 9 y}{(x + y) (- 1 (- y + x))}$

Step 13.5

Reorder terms.

$- \frac{2 (- 5 x - 2 y)}{(x + y) (x - y)} - \frac{7 x - 9 y}{(x + y) (- 1 (x - y))}$

Step 14

To write $- \frac{2 (- 5 x - 2 y)}{(x + y) (x - y)}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$- \frac{2 (- 5 x - 2 y)}{(x + y) (x - y)} \cdot \frac{- 1}{- 1} - \frac{7 x - 9 y}{(x + y) (- 1 (x - y))}$

Step 15

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

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Step 15.1

Multiply $\frac{2 (- 5 x - 2 y)}{(x + y) (x - y)}$ by $\frac{- 1}{- 1}$ .

$- \frac{2 (- 5 x - 2 y) \cdot - 1}{(x + y) (x - y) \cdot - 1} - \frac{7 x - 9 y}{(x + y) (- 1 (x - y))}$

Step 15.2

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

$- \frac{2 (- 5 x - 2 y) \cdot - 1}{- (x + y) (x - y)} - \frac{7 x - 9 y}{(x + y) (- 1 (x - y))}$

Step 15.3

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

$- \frac{2 (- 5 x - 2 y) \cdot - 1}{- (x + y) (x - y)} - \frac{7 x - 9 y}{- (x + y) (x - y)}$

Step 16

Combine the numerators over the common denominator.

$\frac{- 2 (- 5 x - 2 y) \cdot - 1 - (7 x - 9 y)}{- (x + y) (x - y)}$

Step 17

Simplify the numerator.

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Step 17.1

Factor $- 1$ out of $- 2 (- 5 x - 2 y) \cdot - 1 - (7 x - 9 y)$ .

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Step 17.1.1

Move $- 5 x - 2 y$ .

$\frac{- 2 \cdot - 1 (- 5 x - 2 y) - (7 x - 9 y)}{- (x + y) (x - y)}$

Step 17.1.2

Factor $- 1$ out of $- 2 \cdot - 1 (- 5 x - 2 y)$ .

$\frac{- (- 2 (- 5 x - 2 y)) - (7 x - 9 y)}{- (x + y) (x - y)}$

Step 17.1.3

Factor $- 1$ out of $- (- 2 (- 5 x - 2 y)) - (7 x - 9 y)$ .

$\frac{- (- 2 (- 5 x - 2 y) + 7 x - 9 y)}{- (x + y) (x - y)}$

Step 17.2

Apply the distributive property.

$\frac{- (- 2 (- 5 x) - 2 (- 2 y) + 7 x - 9 y)}{- (x + y) (x - y)}$

Step 17.3

Multiply $- 5$ by $- 2$ .

$\frac{- (10 x - 2 (- 2 y) + 7 x - 9 y)}{- (x + y) (x - y)}$

Step 17.4

Multiply $- 2$ by $- 2$ .

$\frac{- (10 x + 4 y + 7 x - 9 y)}{- (x + y) (x - y)}$

Step 17.5

Add $10 x$ and $7 x$ .

$\frac{- (17 x + 4 y - 9 y)}{- (x + y) (x - y)}$

Step 17.6

Subtract $9 y$ from $4 y$ .

$\frac{- (17 x - 5 y)}{- (x + y) (x - y)}$

Step 18

Dividing two negative values results in a positive value.

$\frac{17 x - 5 y}{(x + y) (x - y)}$