Pre-Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Factor $y$ out of $x y$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.2

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

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Step 1.2.1

Factor $x$ out of $x y$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 2

Simplify with factoring out.

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Step 2.1

Factor $- 1$ out of $y$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Simplify the expression.

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Step 2.4.1

Move a negative from the denominator of $\frac{2 x - y}{x (- 1 (- y + x))}$ to the numerator.

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

Step 2.4.2

Reorder terms.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $x$ by $x$ .

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Multiply $2$ by $- 1$ .

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

Step 7.4

Multiply $- - y$ .

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Step 7.4.1

Multiply $- 1$ by $- 1$ .

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

Step 7.4.2

Multiply $y$ by $1$ .

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

Step 7.5

Apply the distributive property.

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

Step 7.6

Multiply $y$ by $y$ .

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

Step 7.7

Factor using the perfect square rule.

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Step 7.7.1

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x y = 2 \cdot x \cdot y$

Step 7.7.2

Rewrite the polynomial.

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

Step 7.7.3

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = y$ .

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

Step 8

Cancel the common factor of ${(x - y)}^{2}$ and $x - y$ .

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Step 8.1

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

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

Step 8.2

Cancel the common factors.

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Step 8.2.1

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

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

Step 8.2.2

Cancel the common factor.

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

Step 8.2.3

Rewrite the expression.

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