Simplify 5/(y-3)+1/(y+3)-(4y-18)/(y^2-9)

$\frac{5}{y - 3} + \frac{1}{y + 3} - \frac{4 y - 18}{y^{2} - 9}$

Step 1

Simplify each term.

Factor $2$ out of $4 y - 18$ .

Factor $2$ out of $4 y$ .

$\frac{5}{y - 3} + \frac{1}{y + 3} - \frac{2 (2 y) - 18}{y^{2} - 9}$

Factor $2$ out of $- 18$ .

$\frac{5}{y - 3} + \frac{1}{y + 3} - \frac{2 (2 y) + 2 (- 9)}{y^{2} - 9}$

Factor $2$ out of $2 (2 y) + 2 (- 9)$ .

$\frac{5}{y - 3} + \frac{1}{y + 3} - \frac{2 (2 y - 9)}{y^{2} - 9}$

Simplify the denominator.

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

$\frac{5}{y - 3} + \frac{1}{y + 3} - \frac{2 (2 y - 9)}{y^{2} - 3^{2}}$

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

$\frac{5}{y - 3} + \frac{1}{y + 3} - \frac{2 (2 y - 9)}{(y + 3) (y - 3)}$

Step 2

To write $\frac{5}{y - 3}$ as a fraction with a common denominator, multiply by $\frac{y + 3}{y + 3}$ .

$\frac{5}{y - 3} \cdot \frac{y + 3}{y + 3} + \frac{1}{y + 3} - \frac{2 (2 y - 9)}{(y + 3) (y - 3)}$

Step 3

To write $\frac{1}{y + 3}$ as a fraction with a common denominator, multiply by $\frac{y - 3}{y - 3}$ .

$\frac{5}{y - 3} \cdot \frac{y + 3}{y + 3} + \frac{1}{y + 3} \cdot \frac{y - 3}{y - 3} - \frac{2 (2 y - 9)}{(y + 3) (y - 3)}$

Step 4

Write each expression with a common denominator of $(y - 3) (y + 3)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5}{y - 3}$ by $\frac{y + 3}{y + 3}$ .

$\frac{5 (y + 3)}{(y - 3) (y + 3)} + \frac{1}{y + 3} \cdot \frac{y - 3}{y - 3} - \frac{2 (2 y - 9)}{(y + 3) (y - 3)}$

Multiply $\frac{1}{y + 3}$ by $\frac{y - 3}{y - 3}$ .

$\frac{5 (y + 3)}{(y - 3) (y + 3)} + \frac{y - 3}{(y + 3) (y - 3)} - \frac{2 (2 y - 9)}{(y + 3) (y - 3)}$

Reorder the factors of $(y - 3) (y + 3)$ .

$\frac{5 (y + 3)}{(y + 3) (y - 3)} + \frac{y - 3}{(y + 3) (y - 3)} - \frac{2 (2 y - 9)}{(y + 3) (y - 3)}$

Step 5

Simplify terms.

Combine the numerators over the common denominator.

$\frac{5 (y + 3) + y - 3}{(y + 3) (y - 3)} - \frac{2 (2 y - 9)}{(y + 3) (y - 3)}$

Combine the numerators over the common denominator.

$\frac{5 (y + 3) + y - 3 - 2 (2 y - 9)}{(y + 3) (y - 3)}$

Step 6

Simplify each term.

Apply the distributive property.

$\frac{5 y + 5 \cdot 3 + y - 3 - 2 (2 y - 9)}{(y + 3) (y - 3)}$

Multiply $5$ by $3$ .

$\frac{5 y + 15 + y - 3 - 2 (2 y - 9)}{(y + 3) (y - 3)}$

Apply the distributive property.

$\frac{5 y + 15 + y - 3 - 2 (2 y) - 2 \cdot - 9}{(y + 3) (y - 3)}$

Multiply $2$ by $- 2$ .

$\frac{5 y + 15 + y - 3 - 4 y - 2 \cdot - 9}{(y + 3) (y - 3)}$

Multiply $- 2$ by $- 9$ .

$\frac{5 y + 15 + y - 3 - 4 y + 18}{(y + 3) (y - 3)}$

Step 7

Simplify terms.

Add $5 y$ and $y$ .

$\frac{6 y + 15 - 3 - 4 y + 18}{(y + 3) (y - 3)}$

Subtract $4 y$ from $6 y$ .

$\frac{2 y + 15 - 3 + 18}{(y + 3) (y - 3)}$

Simplify by adding and subtracting.

Subtract $3$ from $15$ .

$\frac{2 y + 12 + 18}{(y + 3) (y - 3)}$

Add $12$ and $18$ .

$\frac{2 y + 30}{(y + 3) (y - 3)}$

Factor $2$ out of $2 y + 30$ .

Factor $2$ out of $2 y$ .

$\frac{2 (y) + 30}{(y + 3) (y - 3)}$

Factor $2$ out of $30$ .

$\frac{2 y + 2 \cdot 15}{(y + 3) (y - 3)}$

Factor $2$ out of $2 y + 2 \cdot 15$ .

$\frac{2 (y + 15)}{(y + 3) (y - 3)}$

Simplify 5/(y-3)+1/(y+3)-(4y-18)/(y^2-9)

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