Algebra Examples

$\frac{4}{y + 3} - \frac{6}{y - 3} = \frac{4}{y^{2} - 9}$

Step 1

Subtract $\frac{4}{y^{2} - 9}$ from both sides of the equation.

$\frac{4}{y + 3} - \frac{6}{y - 3} - \frac{4}{y^{2} - 9} = 0$

Step 2

Simplify $\frac{4}{y + 3} - \frac{6}{y - 3} - \frac{4}{y^{2} - 9}$ .

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Step 2.1

Simplify the denominator.

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Step 2.1.1

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

$\frac{4}{y + 3} - \frac{6}{y - 3} - \frac{4}{y^{2} - 3^{2}} = 0$

Step 2.1.2

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 $b = 3$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Simplify each term.

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Step 2.7.1

Apply the distributive property.

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

Step 2.7.2

Multiply $4$ by $- 3$ .

$\frac{4 y - 12 - 6 (y + 3) - 4}{(y + 3) (y - 3)} = 0$

Step 2.7.3

Apply the distributive property.

$\frac{4 y - 12 - 6 y - 6 \cdot 3 - 4}{(y + 3) (y - 3)} = 0$

Step 2.7.4

Multiply $- 6$ by $3$ .

$\frac{4 y - 12 - 6 y - 18 - 4}{(y + 3) (y - 3)} = 0$

Step 2.8

Subtract $6 y$ from $4 y$ .

$\frac{- 2 y - 12 - 18 - 4}{(y + 3) (y - 3)} = 0$

Step 2.9

Subtract $18$ from $- 12$ .

$\frac{- 2 y - 30 - 4}{(y + 3) (y - 3)} = 0$

Step 2.10

Subtract $4$ from $- 30$ .

$\frac{- 2 y - 34}{(y + 3) (y - 3)} = 0$

Step 2.11

Factor $2$ out of $- 2 y - 34$ .

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Step 2.11.1

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

$\frac{2 (- y) - 34}{(y + 3) (y - 3)} = 0$

Step 2.11.2

Factor $2$ out of $- 34$ .

$\frac{2 (- y) + 2 (- 17)}{(y + 3) (y - 3)} = 0$

Step 2.11.3

Factor $2$ out of $2 (- y) + 2 (- 17)$ .

$\frac{2 (- y - 17)}{(y + 3) (y - 3)} = 0$

Step 2.12

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

$\frac{2 (- (y) - 17)}{(y + 3) (y - 3)} = 0$

Step 2.13

Rewrite $- 17$ as $- 1 (17)$ .

$\frac{2 (- (y) - 1 (17))}{(y + 3) (y - 3)} = 0$

Step 2.14

Factor $- 1$ out of $- (y) - 1 (17)$ .

$\frac{2 (- (y + 17))}{(y + 3) (y - 3)} = 0$

Step 2.15

Rewrite $- (y + 17)$ as $- 1 (y + 17)$ .

$\frac{2 (- 1 (y + 17))}{(y + 3) (y - 3)} = 0$

Step 2.16

Move the negative in front of the fraction.

$- \frac{2 (y + 17)}{(y + 3) (y - 3)} = 0$

Step 3

Set the numerator equal to zero.

$2 (y + 17) = 0$

Step 4

Solve the equation for $y$ .

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Step 4.1

Divide each term in $2 (y + 17) = 0$ by $2$ and simplify.

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Step 4.1.1

Divide each term in $2 (y + 17) = 0$ by $2$ .

$\frac{2 (y + 17)}{2} = \frac{0}{2}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $2$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{2 (y + 17)}{2} = \frac{0}{2}$

Step 4.1.2.1.2

Divide $y + 17$ by $1$ .

$y + 17 = \frac{0}{2}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $2$ .

$y + 17 = 0$

Step 4.2

Subtract $17$ from both sides of the equation.

$y = - 17$