Solve by Factoring 5/(y+5)-9/(y-5)=10/(y^2-25)

$\frac{5}{y + 5} - \frac{9}{y - 5} = \frac{10}{y^{2} - 25}$

Step 1

Subtract $\frac{10}{y^{2} - 25}$ from both sides of the equation.

$\frac{5}{y + 5} - \frac{9}{y - 5} - \frac{10}{y^{2} - 25} = 0$

Step 2

Simplify $\frac{5}{y + 5} - \frac{9}{y - 5} - \frac{10}{y^{2} - 25}$ .

Simplify the denominator.

Rewrite $25$ as $5^{2}$ .

$\frac{5}{y + 5} - \frac{9}{y - 5} - \frac{10}{y^{2} - 5^{2}} = 0$

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 $b = 5$ .

$\frac{5}{y + 5} - \frac{9}{y - 5} - \frac{10}{(y + 5) (y - 5)} = 0$

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

$\frac{5}{y + 5} \cdot \frac{y - 5}{y - 5} - \frac{9}{y - 5} - \frac{10}{(y + 5) (y - 5)} = 0$

To write $- \frac{9}{y - 5}$ as a fraction with a common denominator, multiply by $\frac{y + 5}{y + 5}$ .

$\frac{5}{y + 5} \cdot \frac{y - 5}{y - 5} - \frac{9}{y - 5} \cdot \frac{y + 5}{y + 5} - \frac{10}{(y + 5) (y - 5)} = 0$

Write each expression with a common denominator of $(y + 5) (y - 5)$ , by multiplying each by an appropriate factor of $1$ .

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

$\frac{5 (y - 5)}{(y + 5) (y - 5)} - \frac{9}{y - 5} \cdot \frac{y + 5}{y + 5} - \frac{10}{(y + 5) (y - 5)} = 0$

Multiply $\frac{9}{y - 5}$ by $\frac{y + 5}{y + 5}$ .

$\frac{5 (y - 5)}{(y + 5) (y - 5)} - \frac{9 (y + 5)}{(y - 5) (y + 5)} - \frac{10}{(y + 5) (y - 5)} = 0$

Reorder the factors of $(y - 5) (y + 5)$ .

$\frac{5 (y - 5)}{(y + 5) (y - 5)} - \frac{9 (y + 5)}{(y + 5) (y - 5)} - \frac{10}{(y + 5) (y - 5)} = 0$

Combine the numerators over the common denominator.

$\frac{5 (y - 5) - 9 (y + 5)}{(y + 5) (y - 5)} - \frac{10}{(y + 5) (y - 5)} = 0$

Combine the numerators over the common denominator.

$\frac{5 (y - 5) - 9 (y + 5) - 10}{(y + 5) (y - 5)} = 0$

Simplify each term.

Apply the distributive property.

$\frac{5 y + 5 \cdot - 5 - 9 (y + 5) - 10}{(y + 5) (y - 5)} = 0$

Multiply $5$ by $- 5$ .

$\frac{5 y - 25 - 9 (y + 5) - 10}{(y + 5) (y - 5)} = 0$

Apply the distributive property.

$\frac{5 y - 25 - 9 y - 9 \cdot 5 - 10}{(y + 5) (y - 5)} = 0$

Multiply $- 9$ by $5$ .

$\frac{5 y - 25 - 9 y - 45 - 10}{(y + 5) (y - 5)} = 0$

Subtract $9 y$ from $5 y$ .

$\frac{- 4 y - 25 - 45 - 10}{(y + 5) (y - 5)} = 0$

Subtract $45$ from $- 25$ .

$\frac{- 4 y - 70 - 10}{(y + 5) (y - 5)} = 0$

Subtract $10$ from $- 70$ .

$\frac{- 4 y - 80}{(y + 5) (y - 5)} = 0$

Factor $4$ out of $- 4 y - 80$ .

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

$\frac{4 (- y) - 80}{(y + 5) (y - 5)} = 0$

Factor $4$ out of $- 80$ .

$\frac{4 (- y) + 4 (- 20)}{(y + 5) (y - 5)} = 0$

Factor $4$ out of $4 (- y) + 4 (- 20)$ .

$\frac{4 (- y - 20)}{(y + 5) (y - 5)} = 0$

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

$\frac{4 (- (y) - 20)}{(y + 5) (y - 5)} = 0$

Rewrite $- 20$ as $- 1 (20)$ .

$\frac{4 (- (y) - 1 (20))}{(y + 5) (y - 5)} = 0$

Factor $- 1$ out of $- (y) - 1 (20)$ .

$\frac{4 (- (y + 20))}{(y + 5) (y - 5)} = 0$

Rewrite $- (y + 20)$ as $- 1 (y + 20)$ .

$\frac{4 (- 1 (y + 20))}{(y + 5) (y - 5)} = 0$

Move the negative in front of the fraction.

$- \frac{4 (y + 20)}{(y + 5) (y - 5)} = 0$

Step 3

Set the numerator equal to zero.

$4 (y + 20) = 0$

Step 4

Solve the equation for $y$ .

Divide each term in $4 (y + 20) = 0$ by $4$ and simplify.

Divide each term in $4 (y + 20) = 0$ by $4$ .

$\frac{4 (y + 20)}{4} = \frac{0}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 (y + 20)}{4} = \frac{0}{4}$

Divide $y + 20$ by $1$ .

$y + 20 = \frac{0}{4}$

Simplify the right side.

Divide $0$ by $4$ .

$y + 20 = 0$

Subtract $20$ from both sides of the equation.

$y = - 20$

Solve by Factoring 5/(y+5)-9/(y-5)=10/(y^2-25)

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