Algebra Examples

$\frac{2 x + 2 y}{x^{2} - 2 x y + y^{2}} \cdot \frac{x^{2} - x y}{7 y + 7 x}$

Step 1

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

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Step 1.1

Factor $2$ out of $2 x$ .

$\frac{2 (x) + 2 y}{x^{2} - 2 x y + y^{2}} \cdot \frac{x^{2} - x y}{7 y + 7 x}$

Step 1.2

Factor $2$ out of $2 y$ .

$\frac{2 (x) + 2 (y)}{x^{2} - 2 x y + y^{2}} \cdot \frac{x^{2} - x y}{7 y + 7 x}$

Step 1.3

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

$\frac{2 (x + y)}{x^{2} - 2 x y + y^{2}} \cdot \frac{x^{2} - x y}{7 y + 7 x}$

Step 2

Factor using the perfect square rule.

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Step 2.1

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 2.2

Rewrite the polynomial.

$\frac{2 (x + y)}{x^{2} - 2 \cdot x \cdot y + y^{2}} \cdot \frac{x^{2} - x y}{7 y + 7 x}$

Step 2.3

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

$\frac{2 (x + y)}{{(x - y)}^{2}} \cdot \frac{x^{2} - x y}{7 y + 7 x}$

Step 3

Simplify the numerator.

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Step 3.1

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

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Step 3.1.1

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

$\frac{2 (x + y)}{{(x - y)}^{2}} \cdot \frac{x \cdot x - x y}{7 y + 7 x}$

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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

$\frac{2 (x + y)}{{(x - y)}^{2}} \cdot \frac{x (x - y)}{7 y + 7 x}$

Step 4

Simplify terms.

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Step 4.1

Factor $7$ out of $7 y + 7 x$ .

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Step 4.1.1

Factor $7$ out of $7 y$ .

$\frac{2 (x + y)}{{(x - y)}^{2}} \cdot \frac{x (x - y)}{7 (y) + 7 x}$

Step 4.1.2

Factor $7$ out of $7 x$ .

$\frac{2 (x + y)}{{(x - y)}^{2}} \cdot \frac{x (x - y)}{7 (y) + 7 (x)}$

Step 4.1.3

Factor $7$ out of $7 (y) + 7 (x)$ .

$\frac{2 (x + y)}{{(x - y)}^{2}} \cdot \frac{x (x - y)}{7 (y + x)}$

Step 4.2

Combine.

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $x - y$ out of ${(x - y)}^{2} (7 (y + x))$ .

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

Step 4.3.2.2

Cancel the common factor.

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

Step 4.3.2.3

Rewrite the expression.

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

Step 4.4

Cancel the common factor of $x + y$ and $y + x$ .

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Step 4.4.1

Reorder terms.

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

Step 4.4.2

Cancel the common factor.

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

Step 4.4.3

Rewrite the expression.

$\frac{2 (x)}{(x - y) \cdot (7)}$

Step 4.5

Move $7$ to the left of $x - y$ .

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