Solve by Factoring (y+1)/(y+6)-y/(y^2-36)=(y-3)/(y-6)

$\frac{y + 1}{y + 6} - \frac{y}{y^{2} - 36} = \frac{y - 3}{y - 6}$

Step 1

Subtract $\frac{y - 3}{y - 6}$ from both sides of the equation.

$\frac{y + 1}{y + 6} - \frac{y}{y^{2} - 36} - \frac{y - 3}{y - 6} = 0$

Step 2

Simplify $\frac{y + 1}{y + 6} - \frac{y}{y^{2} - 36} - \frac{y - 3}{y - 6}$ .

Simplify the denominator.

Rewrite $36$ as $6^{2}$ .

$\frac{y + 1}{y + 6} - \frac{y}{y^{2} - 6^{2}} - \frac{y - 3}{y - 6} = 0$

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 $b = 6$ .

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

To write $\frac{y + 1}{y + 6}$ as a fraction with a common denominator, multiply by $\frac{y - 6}{y - 6}$ .

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

Multiply $\frac{y + 1}{y + 6}$ by $\frac{y - 6}{y - 6}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Expand $(y + 1) (y - 6)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

$\frac{y^{2} + y \cdot - 6 + 1 y + 1 \cdot - 6 - y}{(y + 6) (y - 6)} - \frac{y - 3}{y - 6} = 0$

Move $- 6$ to the left of $y$ .

$\frac{y^{2} - 6 \cdot y + 1 y + 1 \cdot - 6 - y}{(y + 6) (y - 6)} - \frac{y - 3}{y - 6} = 0$

Multiply $y$ by $1$ .

$\frac{y^{2} - 6 y + y + 1 \cdot - 6 - y}{(y + 6) (y - 6)} - \frac{y - 3}{y - 6} = 0$

Multiply $- 6$ by $1$ .

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

Add $- 6 y$ and $y$ .

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

Subtract $y$ from $- 5 y$ .

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

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

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

Write each expression with a common denominator of $(y + 6) (y - 6)$ , by multiplying each by an appropriate factor of $1$ .

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

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

Reorder the factors of $(y - 6) (y + 6)$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 1$ by $- 3$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $y$ .

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

Multiply $y$ by $y$ .

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

Multiply $6$ by $- 1$ .

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

Multiply $3$ by $6$ .

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

Add $- 6 y$ and $3 y$ .

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

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

$\frac{- 6 y - 6 + 0 - 3 y + 18}{(y + 6) (y - 6)} = 0$

Add $- 6 y$ and $0$ .

$\frac{- 6 y - 6 - 3 y + 18}{(y + 6) (y - 6)} = 0$

Subtract $3 y$ from $- 6 y$ .

$\frac{- 9 y - 6 + 18}{(y + 6) (y - 6)} = 0$

Add $- 6$ and $18$ .

$\frac{- 9 y + 12}{(y + 6) (y - 6)} = 0$

Factor $3$ out of $- 9 y + 12$ .

Factor $3$ out of $- 9 y$ .

$\frac{3 (- 3 y) + 12}{(y + 6) (y - 6)} = 0$

Factor $3$ out of $12$ .

$\frac{3 (- 3 y) + 3 (4)}{(y + 6) (y - 6)} = 0$

Factor $3$ out of $3 (- 3 y) + 3 (4)$ .

$\frac{3 (- 3 y + 4)}{(y + 6) (y - 6)} = 0$

Factor $- 1$ out of $- 3 y$ .

$\frac{3 (- (3 y) + 4)}{(y + 6) (y - 6)} = 0$

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{3 (- (3 y) - 1 (- 4))}{(y + 6) (y - 6)} = 0$

Factor $- 1$ out of $- (3 y) - 1 (- 4)$ .

$\frac{3 (- (3 y - 4))}{(y + 6) (y - 6)} = 0$

Rewrite $- (3 y - 4)$ as $- 1 (3 y - 4)$ .

$\frac{3 (- 1 (3 y - 4))}{(y + 6) (y - 6)} = 0$

Move the negative in front of the fraction.

$- \frac{3 (3 y - 4)}{(y + 6) (y - 6)} = 0$

Step 3

Set the numerator equal to zero.

$3 (3 y - 4) = 0$

Step 4

Solve the equation for $y$ .

Divide each term in $3 (3 y - 4) = 0$ by $3$ and simplify.

Divide each term in $3 (3 y - 4) = 0$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $3 y - 4$ by $1$ .

$3 y - 4 = \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$3 y - 4 = 0$

Add $4$ to both sides of the equation.

$3 y = 4$

Divide each term in $3 y = 4$ by $3$ and simplify.

Divide each term in $3 y = 4$ by $3$ .

$\frac{3 y}{3} = \frac{4}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y}{3} = \frac{4}{3}$

Divide $y$ by $1$ .

$y = \frac{4}{3}$

Solve by Factoring (y+1)/(y+6)-y/(y^2-36)=(y-3)/(y-6)

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