Algebra Examples

$\frac{x^{2} - 3 x y}{(x + y) (x - y)} + \frac{y}{x - y}$

Step 1

Factor $x$ out of $x^{2} - 3 x y$ .

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Step 1.1

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

$\frac{x \cdot x - 3 x y}{(x + y) (x - y)} + \frac{y}{x - y}$

Step 1.2

Factor $x$ out of $- 3 x y$ .

$\frac{x \cdot x + x (- 3 y)}{(x + y) (x - y)} + \frac{y}{x - y}$

Step 1.3

Factor $x$ out of $x \cdot x + x (- 3 y)$ .

$\frac{x (x - 3 y)}{(x + y) (x - y)} + \frac{y}{x - y}$

Step 2

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

$\frac{x (x - 3 y)}{(x + y) (x - y)} + \frac{y}{x - y} \cdot \frac{x + y}{x + y}$

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

$\frac{x (x - 3 y)}{(x + y) (x - y)} + \frac{y (x + y)}{(x - y) (x + y)}$

Step 3.2

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

$\frac{x (x - 3 y)}{(x + y) (x - y)} + \frac{y (x + y)}{(x + y) (x - y)}$

Step 4

Combine the numerators over the common denominator.

$\frac{x (x - 3 y) + y (x + y)}{(x + y) (x - y)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{x \cdot x + x (- 3 y) + y (x + y)}{(x + y) (x - y)}$

Step 5.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x (- 3 y) + y (x + y)}{(x + y) (x - y)}$

Step 5.3

Rewrite using the commutative property of multiplication.

$\frac{x^{2} - 3 x y + y (x + y)}{(x + y) (x - y)}$

Step 5.4

Apply the distributive property.

$\frac{x^{2} - 3 x y + y x + y \cdot y}{(x + y) (x - y)}$

Step 5.5

Multiply $y$ by $y$ .

$\frac{x^{2} - 3 x y + y x + y^{2}}{(x + y) (x - y)}$

Step 5.6

Add $- 3 x y$ and $y x$ .

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Step 5.6.1

Reorder $y$ and $x$ .

$\frac{x^{2} - 3 x y + x y + y^{2}}{(x + y) (x - y)}$

Step 5.6.2

Add $- 3 x y$ and $x y$ .

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

Step 5.7

Factor using the perfect square rule.

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Step 5.7.1

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 5.7.2

Rewrite the polynomial.

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

Step 5.7.3

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 6

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

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Step 6.1

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

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

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

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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