Solve by Factoring 6/(y+3)+2/y=(5y-3)/(y^2-9)

$\frac{6}{y + 3} + \frac{2}{y} = \frac{5 y - 3}{y^{2} - 9}$

Step 1

Subtract $\frac{5 y - 3}{y^{2} - 9}$ from both sides of the equation.

$\frac{6}{y + 3} + \frac{2}{y} - \frac{5 y - 3}{y^{2} - 9} = 0$

Step 2

Simplify $\frac{6}{y + 3} + \frac{2}{y} - \frac{5 y - 3}{y^{2} - 9}$ .

Simplify the denominator.

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

$\frac{6}{y + 3} + \frac{2}{y} - \frac{5 y - 3}{y^{2} - 3^{2}} = 0$

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 $b = 3$ .

$\frac{6}{y + 3} + \frac{2}{y} - \frac{5 y - 3}{(y + 3) (y - 3)} = 0$

To write $\frac{6}{y + 3}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{6}{y + 3} \cdot \frac{y}{y} + \frac{2}{y} - \frac{5 y - 3}{(y + 3) (y - 3)} = 0$

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

$\frac{6}{y + 3} \cdot \frac{y}{y} + \frac{2}{y} \cdot \frac{y + 3}{y + 3} - \frac{5 y - 3}{(y + 3) (y - 3)} = 0$

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

Multiply $\frac{6}{y + 3}$ by $\frac{y}{y}$ .

$\frac{6 y}{(y + 3) y} + \frac{2}{y} \cdot \frac{y + 3}{y + 3} - \frac{5 y - 3}{(y + 3) (y - 3)} = 0$

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

$\frac{6 y}{(y + 3) y} + \frac{2 (y + 3)}{y (y + 3)} - \frac{5 y - 3}{(y + 3) (y - 3)} = 0$

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

$\frac{6 y}{y (y + 3)} + \frac{2 (y + 3)}{y (y + 3)} - \frac{5 y - 3}{(y + 3) (y - 3)} = 0$

Combine the numerators over the common denominator.

$\frac{6 y + 2 (y + 3)}{y (y + 3)} - \frac{5 y - 3}{(y + 3) (y - 3)} = 0$

Simplify the numerator.

Factor $2$ out of $6 y + 2 (y + 3)$ .

Factor $2$ out of $6 y$ .

$\frac{2 (3 y) + 2 (y + 3)}{y (y + 3)} - \frac{5 y - 3}{(y + 3) (y - 3)} = 0$

Factor $2$ out of $2 (3 y) + 2 (y + 3)$ .

$\frac{2 (3 y + y + 3)}{y (y + 3)} - \frac{5 y - 3}{(y + 3) (y - 3)} = 0$

Add $3 y$ and $y$ .

$\frac{2 (4 y + 3)}{y (y + 3)} - \frac{5 y - 3}{(y + 3) (y - 3)} = 0$

To write $\frac{2 (4 y + 3)}{y (y + 3)}$ as a fraction with a common denominator, multiply by $\frac{y - 3}{y - 3}$ .

$\frac{2 (4 y + 3)}{y (y + 3)} \cdot \frac{y - 3}{y - 3} - \frac{5 y - 3}{(y + 3) (y - 3)} = 0$

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

$\frac{2 (4 y + 3)}{y (y + 3)} \cdot \frac{y - 3}{y - 3} - \frac{5 y - 3}{(y + 3) (y - 3)} \cdot \frac{y}{y} = 0$

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

Multiply $\frac{2 (4 y + 3)}{y (y + 3)}$ by $\frac{y - 3}{y - 3}$ .

$\frac{2 (4 y + 3) (y - 3)}{y (y + 3) (y - 3)} - \frac{5 y - 3}{(y + 3) (y - 3)} \cdot \frac{y}{y} = 0$

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

$\frac{2 (4 y + 3) (y - 3)}{y (y + 3) (y - 3)} - \frac{(5 y - 3) y}{(y + 3) (y - 3) y} = 0$

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

$\frac{2 (4 y + 3) (y - 3)}{y (y + 3) (y - 3)} - \frac{(5 y - 3) y}{y (y + 3) (y - 3)} = 0$

Combine the numerators over the common denominator.

$\frac{2 (4 y + 3) (y - 3) - (5 y - 3) y}{y (y + 3) (y - 3)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{(2 (4 y) + 2 \cdot 3) (y - 3) - (5 y - 3) y}{y (y + 3) (y - 3)} = 0$

Multiply $4$ by $2$ .

$\frac{(8 y + 2 \cdot 3) (y - 3) - (5 y - 3) y}{y (y + 3) (y - 3)} = 0$

Multiply $2$ by $3$ .

$\frac{(8 y + 6) (y - 3) - (5 y - 3) y}{y (y + 3) (y - 3)} = 0$

Expand $(8 y + 6) (y - 3)$ using the FOIL Method.

Apply the distributive property.

$\frac{8 y (y - 3) + 6 (y - 3) - (5 y - 3) y}{y (y + 3) (y - 3)} = 0$

Apply the distributive property.

$\frac{8 y \cdot y + 8 y \cdot - 3 + 6 (y - 3) - (5 y - 3) y}{y (y + 3) (y - 3)} = 0$

Apply the distributive property.

$\frac{8 y \cdot y + 8 y \cdot - 3 + 6 y + 6 \cdot - 3 - (5 y - 3) y}{y (y + 3) (y - 3)} = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$\frac{8 (y \cdot y) + 8 y \cdot - 3 + 6 y + 6 \cdot - 3 - (5 y - 3) y}{y (y + 3) (y - 3)} = 0$

Multiply $y$ by $y$ .

$\frac{8 y^{2} + 8 y \cdot - 3 + 6 y + 6 \cdot - 3 - (5 y - 3) y}{y (y + 3) (y - 3)} = 0$

Multiply $- 3$ by $8$ .

$\frac{8 y^{2} - 24 y + 6 y + 6 \cdot - 3 - (5 y - 3) y}{y (y + 3) (y - 3)} = 0$

Multiply $6$ by $- 3$ .

$\frac{8 y^{2} - 24 y + 6 y - 18 - (5 y - 3) y}{y (y + 3) (y - 3)} = 0$

Add $- 24 y$ and $6 y$ .

$\frac{8 y^{2} - 18 y - 18 - (5 y - 3) y}{y (y + 3) (y - 3)} = 0$

Apply the distributive property.

$\frac{8 y^{2} - 18 y - 18 + (- (5 y) - - 3) y}{y (y + 3) (y - 3)} = 0$

Multiply $5$ by $- 1$ .

$\frac{8 y^{2} - 18 y - 18 + (- 5 y - - 3) y}{y (y + 3) (y - 3)} = 0$

Multiply $- 1$ by $- 3$ .

$\frac{8 y^{2} - 18 y - 18 + (- 5 y + 3) y}{y (y + 3) (y - 3)} = 0$

Apply the distributive property.

$\frac{8 y^{2} - 18 y - 18 - 5 y \cdot y + 3 y}{y (y + 3) (y - 3)} = 0$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$\frac{8 y^{2} - 18 y - 18 - 5 (y \cdot y) + 3 y}{y (y + 3) (y - 3)} = 0$

Multiply $y$ by $y$ .

$\frac{8 y^{2} - 18 y - 18 - 5 y^{2} + 3 y}{y (y + 3) (y - 3)} = 0$

Subtract $5 y^{2}$ from $8 y^{2}$ .

$\frac{3 y^{2} - 18 y - 18 + 3 y}{y (y + 3) (y - 3)} = 0$

Add $- 18 y$ and $3 y$ .

$\frac{3 y^{2} - 15 y - 18}{y (y + 3) (y - 3)} = 0$

Rewrite $3 y^{2} - 15 y - 18$ in a factored form.

Factor $3$ out of $3 y^{2} - 15 y - 18$ .

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

$\frac{3 (y^{2}) - 15 y - 18}{y (y + 3) (y - 3)} = 0$

Factor $3$ out of $- 15 y$ .

$\frac{3 (y^{2}) + 3 (- 5 y) - 18}{y (y + 3) (y - 3)} = 0$

Factor $3$ out of $- 18$ .

$\frac{3 y^{2} + 3 (- 5 y) + 3 \cdot - 6}{y (y + 3) (y - 3)} = 0$

Factor $3$ out of $3 y^{2} + 3 (- 5 y)$ .

$\frac{3 (y^{2} - 5 y) + 3 \cdot - 6}{y (y + 3) (y - 3)} = 0$

Factor $3$ out of $3 (y^{2} - 5 y) + 3 \cdot - 6$ .

$\frac{3 (y^{2} - 5 y - 6)}{y (y + 3) (y - 3)} = 0$

Factor $y^{2} - 5 y - 6$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $- 5$ .

$- 6, 1$

Write the factored form using these integers.

$\frac{3 ((y - 6) (y + 1))}{y (y + 3) (y - 3)} = 0$

$\frac{3 (y - 6) (y + 1)}{y (y + 3) (y - 3)} = 0$

Step 3

Set the numerator equal to zero.

$3 (y - 6) (y + 1) = 0$

Step 4

Solve the equation for $y$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y - 6 = 0$

$y + 1 = 0$

Set $y - 6$ equal to $0$ and solve for $y$ .

Set $y - 6$ equal to $0$ .

$y - 6 = 0$

Add $6$ to both sides of the equation.

$y = 6$

Set $y + 1$ equal to $0$ and solve for $y$ .

Set $y + 1$ equal to $0$ .

$y + 1 = 0$

Subtract $1$ from both sides of the equation.

$y = - 1$

The final solution is all the values that make $3 (y - 6) (y + 1) = 0$ true.

$y = 6, - 1$

Solve by Factoring 6/(y+3)+2/y=(5y-3)/(y^2-9)

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