Algebra Examples

$\frac{a^{2} - 4}{x^{2} - 9} \div \frac{a^{2} - 2 a}{x y + 3 y} + \frac{2 - y}{x - 3}$

Step 1

Simplify each term.

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Step 1.1

To divide by a fraction, multiply by its reciprocal.

$\frac{a^{2} - 4}{x^{2} - 9} \cdot \frac{x y + 3 y}{a^{2} - 2 a} + \frac{2 - y}{x - 3}$

Step 1.2

Simplify the numerator.

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Step 1.2.1

Rewrite $4$ as $2^{2}$ .

$\frac{a^{2} - 2^{2}}{x^{2} - 9} \cdot \frac{x y + 3 y}{a^{2} - 2 a} + \frac{2 - y}{x - 3}$

Step 1.2.2

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

$\frac{(a + 2) (a - 2)}{x^{2} - 9} \cdot \frac{x y + 3 y}{a^{2} - 2 a} + \frac{2 - y}{x - 3}$

Step 1.3

Simplify the denominator.

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Step 1.3.1

Rewrite $9$ as $3^{2}$ .

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

Step 1.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

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

Step 1.4

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

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Step 1.4.1

Factor $y$ out of $x y$ .

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

Step 1.4.2

Factor $y$ out of $3 y$ .

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

Step 1.4.3

Factor $y$ out of $y x + y \cdot 3$ .

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

Step 1.5

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

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Step 1.5.1

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

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

Step 1.5.2

Factor $a$ out of $- 2 a$ .

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

Step 1.5.3

Factor $a$ out of $a \cdot a + a \cdot - 2$ .

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

Step 1.6

Cancel the common factor of $a - 2$ .

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Step 1.6.1

Factor $a - 2$ out of $(a + 2) (a - 2)$ .

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

Step 1.6.2

Factor $a - 2$ out of $a (a - 2)$ .

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

Step 1.6.3

Cancel the common factor.

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

Step 1.6.4

Rewrite the expression.

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

Step 1.7

Cancel the common factor of $x + 3$ .

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Step 1.7.1

Factor $x + 3$ out of $y (x + 3)$ .

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

Step 1.7.2

Cancel the common factor.

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

Step 1.7.3

Rewrite the expression.

$\frac{a + 2}{x - 3} \cdot \frac{y}{a} + \frac{2 - y}{x - 3}$

Step 1.8

Multiply $\frac{a + 2}{x - 3}$ by $\frac{y}{a}$ .

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

Step 2

To write $\frac{2 - y}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

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

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{2 - y}{x - 3}$ by $\frac{a}{a}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Subtract $y a$ from $a y$ .

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Step 4.3.1

Move $y$ .

$\frac{2 y + 2 a + a y - 1 a y}{(x - 3) a}$

Step 4.3.2

Subtract $a y$ from $a y$ .

$\frac{2 y + 2 a + 0}{(x - 3) a}$

Step 4.4

Add $2 y + 2 a$ and $0$ .

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

Step 4.5

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

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Step 4.5.1

Factor $2$ out of $2 y$ .

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

Step 4.5.2

Factor $2$ out of $2 a$ .

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

Step 4.5.3

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

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

Step 5

Reorder factors in $\frac{2 (y + a)}{(x - 3) a}$ .

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