Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Factor $y$ out of $2 x y$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.2

Simplify the denominator.

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Step 1.2.1

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

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Step 1.2.1.1

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

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

Step 1.2.1.2

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

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

Step 1.2.1.3

Factor $x$ out of $x (2 x) + x (- 1 y)$ .

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

Step 1.2.2

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

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

Step 2

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

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $y (2 x - y) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $y (2 x - y) x$ .

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

Step 4.4

Reorder the factors of $x (2 x - y) y$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $x$ by $x$ .

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Multiply $- - y$ .

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Step 6.4.1

Multiply $- 1$ by $- 1$ .

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

Step 6.4.2

Multiply $y$ by $1$ .

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

Step 6.5

Apply the distributive property.

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

Step 6.6

Multiply $y$ by $y$ .

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

Step 6.7

Subtract $x y$ from $y x$ .

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Step 6.7.1

Reorder $y$ and $x$ .

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

Step 6.7.2

Subtract $x y$ from $x y$ .

$\frac{x^{2} + 0 + y^{2}}{x y (2 x - y)}$

Step 6.8

Add $x^{2}$ and $0$ .

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