Solve by Factoring 5/(y-3)-30/(y^2-9)=1

$\frac{5}{y - 3} - \frac{30}{y^{2} - 9} = 1$

Step 1

Subtract $1$ from both sides of the equation.

$\frac{5}{y - 3} - \frac{30}{y^{2} - 9} - 1 = 0$

Step 2

Simplify $\frac{5}{y - 3} - \frac{30}{y^{2} - 9} - 1$ .

Simplify the denominator.

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

$\frac{5}{y - 3} - \frac{30}{y^{2} - 3^{2}} - 1 = 0$

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 $b = 3$ .

$\frac{5}{y - 3} - \frac{30}{(y + 3) (y - 3)} - 1 = 0$

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

$\frac{5}{y - 3} \cdot \frac{y + 3}{y + 3} - \frac{30}{(y + 3) (y - 3)} - 1 = 0$

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 $\frac{y + 3}{y + 3}$ .

$\frac{5 (y + 3)}{(y - 3) (y + 3)} - \frac{30}{(y + 3) (y - 3)} - 1 = 0$

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

$\frac{5 (y + 3)}{(y + 3) (y - 3)} - \frac{30}{(y + 3) (y - 3)} - 1 = 0$

Combine the numerators over the common denominator.

$\frac{5 (y + 3) - 30}{(y + 3) (y - 3)} - 1 = 0$

Simplify each term.

Simplify the numerator.

Factor $5$ out of $5 (y + 3) - 30$ .

Factor $5$ out of $- 30$ .

$\frac{5 (y + 3) + 5 \cdot - 6}{(y + 3) (y - 3)} - 1 = 0$

Factor $5$ out of $5 (y + 3) + 5 \cdot - 6$ .

$\frac{5 (y + 3 - 6)}{(y + 3) (y - 3)} - 1 = 0$

Subtract $6$ from $3$ .

$\frac{5 (y - 3)}{(y + 3) (y - 3)} - 1 = 0$

Cancel the common factor of $y - 3$ .

Cancel the common factor.

$\frac{5 (y - 3)}{(y + 3) (y - 3)} - 1 = 0$

Rewrite the expression.

$\frac{5}{y + 3} - 1 = 0$

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

$\frac{5}{y + 3} - 1 \cdot \frac{y + 3}{y + 3} = 0$

Combine $- 1$ and $\frac{y + 3}{y + 3}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

$\frac{5 - y - 1 \cdot 3}{y + 3} = 0$

Multiply $- 1$ by $3$ .

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

Subtract $3$ from $5$ .

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

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

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

Rewrite $2$ as $- 1 (- 2)$ .

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

Factor $- 1$ out of $- (y) - 1 (- 2)$ .

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

Rewrite $- (y - 2)$ as $- 1 (y - 2)$ .

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

Move the negative in front of the fraction.

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

Step 3

Set the numerator equal to zero.

$y - 2 = 0$

Step 4

Add $2$ to both sides of the equation.

$y = 2$

Solve by Factoring 5/(y-3)-30/(y^2-9)=1

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