Algebra Examples

$\frac{10}{2 y + 8} - \frac{7 y + 8}{y^{2} - 16} = \frac{- 8}{2 y - 8}$

Step 1

Subtract $\frac{- 8}{2 y - 8}$ from both sides of the equation.

$\frac{10}{2 y + 8} - \frac{7 y + 8}{y^{2} - 16} - \frac{- 8}{2 y - 8} = 0$

Step 2

Simplify $\frac{10}{2 y + 8} - \frac{7 y + 8}{y^{2} - 16} - \frac{- 8}{2 y - 8}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Cancel the common factor of $10$ and $2 y + 8$ .

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Step 2.1.1.1

Factor $2$ out of $10$ .

$\frac{2 \cdot 5}{2 y + 8} - \frac{7 y + 8}{y^{2} - 16} - \frac{- 8}{2 y - 8} = 0$

Step 2.1.1.2

Cancel the common factors.

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Step 2.1.1.2.1

Factor $2$ out of $2 y$ .

$\frac{2 \cdot 5}{2 (y) + 8} - \frac{7 y + 8}{y^{2} - 16} - \frac{- 8}{2 y - 8} = 0$

Step 2.1.1.2.2

Factor $2$ out of $8$ .

$\frac{2 \cdot 5}{2 (y) + 2 (4)} - \frac{7 y + 8}{y^{2} - 16} - \frac{- 8}{2 y - 8} = 0$

Step 2.1.1.2.3

Factor $2$ out of $2 (y) + 2 (4)$ .

$\frac{2 \cdot 5}{2 (y + 4)} - \frac{7 y + 8}{y^{2} - 16} - \frac{- 8}{2 y - 8} = 0$

Step 2.1.1.2.4

Cancel the common factor.

$\frac{2 \cdot 5}{2 (y + 4)} - \frac{7 y + 8}{y^{2} - 16} - \frac{- 8}{2 y - 8} = 0$

Step 2.1.1.2.5

Rewrite the expression.

$\frac{5}{y + 4} - \frac{7 y + 8}{y^{2} - 16} - \frac{- 8}{2 y - 8} = 0$

Step 2.1.2

Simplify the denominator.

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Step 2.1.2.1

Rewrite $16$ as $4^{2}$ .

$\frac{5}{y + 4} - \frac{7 y + 8}{y^{2} - 4^{2}} - \frac{- 8}{2 y - 8} = 0$

Step 2.1.2.2

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 $b = 4$ .

$\frac{5}{y + 4} - \frac{7 y + 8}{(y + 4) (y - 4)} - \frac{- 8}{2 y - 8} = 0$

Step 2.1.3

Cancel the common factor of $- 8$ and $2 y - 8$ .

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Step 2.1.3.1

Factor $2$ out of $- 8$ .

$\frac{5}{y + 4} - \frac{7 y + 8}{(y + 4) (y - 4)} - \frac{2 \cdot - 4}{2 y - 8} = 0$

Step 2.1.3.2

Cancel the common factors.

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Step 2.1.3.2.1

Factor $2$ out of $2 y$ .

$\frac{5}{y + 4} - \frac{7 y + 8}{(y + 4) (y - 4)} - \frac{2 \cdot - 4}{2 (y) - 8} = 0$

Step 2.1.3.2.2

Factor $2$ out of $- 8$ .

$\frac{5}{y + 4} - \frac{7 y + 8}{(y + 4) (y - 4)} - \frac{2 \cdot - 4}{2 (y) + 2 (- 4)} = 0$

Step 2.1.3.2.3

Factor $2$ out of $2 (y) + 2 (- 4)$ .

$\frac{5}{y + 4} - \frac{7 y + 8}{(y + 4) (y - 4)} - \frac{2 \cdot - 4}{2 (y - 4)} = 0$

Step 2.1.3.2.4

Cancel the common factor.

$\frac{5}{y + 4} - \frac{7 y + 8}{(y + 4) (y - 4)} - \frac{2 \cdot - 4}{2 (y - 4)} = 0$

Step 2.1.3.2.5

Rewrite the expression.

$\frac{5}{y + 4} - \frac{7 y + 8}{(y + 4) (y - 4)} - \frac{- 4}{y - 4} = 0$

Step 2.1.4

Move the negative in front of the fraction.

$\frac{5}{y + 4} - \frac{7 y + 8}{(y + 4) (y - 4)} - - \frac{4}{y - 4} = 0$

Step 2.1.5

Multiply $- - \frac{4}{y - 4}$ .

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Step 2.1.5.1

Multiply $- 1$ by $- 1$ .

$\frac{5}{y + 4} - \frac{7 y + 8}{(y + 4) (y - 4)} + 1 \frac{4}{y - 4} = 0$

Step 2.1.5.2

Multiply $\frac{4}{y - 4}$ by $1$ .

$\frac{5}{y + 4} - \frac{7 y + 8}{(y + 4) (y - 4)} + \frac{4}{y - 4} = 0$

Step 2.2

To write $\frac{5}{y + 4}$ as a fraction with a common denominator, multiply by $\frac{y - 4}{y - 4}$ .

$\frac{5}{y + 4} \cdot \frac{y - 4}{y - 4} - \frac{7 y + 8}{(y + 4) (y - 4)} + \frac{4}{y - 4} = 0$

Step 2.3

Multiply $\frac{5}{y + 4}$ by $\frac{y - 4}{y - 4}$ .

$\frac{5 (y - 4)}{(y + 4) (y - 4)} - \frac{7 y + 8}{(y + 4) (y - 4)} + \frac{4}{y - 4} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{5 (y - 4) - (7 y + 8)}{(y + 4) (y - 4)} + \frac{4}{y - 4} = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

$\frac{5 y + 5 \cdot - 4 - (7 y + 8)}{(y + 4) (y - 4)} + \frac{4}{y - 4} = 0$

Step 2.5.2

Multiply $5$ by $- 4$ .

$\frac{5 y - 20 - (7 y + 8)}{(y + 4) (y - 4)} + \frac{4}{y - 4} = 0$

Step 2.5.3

Apply the distributive property.

$\frac{5 y - 20 - (7 y) - 1 \cdot 8}{(y + 4) (y - 4)} + \frac{4}{y - 4} = 0$

Step 2.5.4

Multiply $7$ by $- 1$ .

$\frac{5 y - 20 - 7 y - 1 \cdot 8}{(y + 4) (y - 4)} + \frac{4}{y - 4} = 0$

Step 2.5.5

Multiply $- 1$ by $8$ .

$\frac{5 y - 20 - 7 y - 8}{(y + 4) (y - 4)} + \frac{4}{y - 4} = 0$

Step 2.5.6

Subtract $7 y$ from $5 y$ .

$\frac{- 2 y - 20 - 8}{(y + 4) (y - 4)} + \frac{4}{y - 4} = 0$

Step 2.5.7

Subtract $8$ from $- 20$ .

$\frac{- 2 y - 28}{(y + 4) (y - 4)} + \frac{4}{y - 4} = 0$

Step 2.5.8

Factor $2$ out of $- 2 y - 28$ .

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Step 2.5.8.1

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

$\frac{2 (- y) - 28}{(y + 4) (y - 4)} + \frac{4}{y - 4} = 0$

Step 2.5.8.2

Factor $2$ out of $- 28$ .

$\frac{2 (- y) + 2 (- 14)}{(y + 4) (y - 4)} + \frac{4}{y - 4} = 0$

Step 2.5.8.3

Factor $2$ out of $2 (- y) + 2 (- 14)$ .

$\frac{2 (- y - 14)}{(y + 4) (y - 4)} + \frac{4}{y - 4} = 0$

Step 2.6

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

$\frac{2 (- y - 14)}{(y + 4) (y - 4)} + \frac{4}{y - 4} \cdot \frac{y + 4}{y + 4} = 0$

Step 2.7

Write each expression with a common denominator of $(y + 4) (y - 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.7.1

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

$\frac{2 (- y - 14)}{(y + 4) (y - 4)} + \frac{4 (y + 4)}{(y - 4) (y + 4)} = 0$

Step 2.7.2

Reorder the factors of $(y - 4) (y + 4)$ .

$\frac{2 (- y - 14)}{(y + 4) (y - 4)} + \frac{4 (y + 4)}{(y + 4) (y - 4)} = 0$

Step 2.8

Combine the numerators over the common denominator.

$\frac{2 (- y - 14) + 4 (y + 4)}{(y + 4) (y - 4)} = 0$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Factor $2$ out of $2 (- y - 14) + 4 (y + 4)$ .

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Step 2.9.1.1

Factor $2$ out of $4 (y + 4)$ .

$\frac{2 (- y - 14) + 2 (2 (y + 4))}{(y + 4) (y - 4)} = 0$

Step 2.9.1.2

Factor $2$ out of $2 (- y - 14) + 2 (2 (y + 4))$ .

$\frac{2 (- y - 14 + 2 (y + 4))}{(y + 4) (y - 4)} = 0$

Step 2.9.2

Apply the distributive property.

$\frac{2 (- y - 14 + 2 y + 2 \cdot 4)}{(y + 4) (y - 4)} = 0$

Step 2.9.3

Multiply $2$ by $4$ .

$\frac{2 (- y - 14 + 2 y + 8)}{(y + 4) (y - 4)} = 0$

Step 2.9.4

Add $- y$ and $2 y$ .

$\frac{2 (y - 14 + 8)}{(y + 4) (y - 4)} = 0$

Step 2.9.5

Add $- 14$ and $8$ .

$\frac{2 (y - 6)}{(y + 4) (y - 4)} = 0$

Step 3

Set the numerator equal to zero.

$2 (y - 6) = 0$

Step 4

Solve the equation for $y$ .

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Step 4.1

Divide each term in $2 (y - 6) = 0$ by $2$ and simplify.

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Step 4.1.1

Divide each term in $2 (y - 6) = 0$ by $2$ .

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

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.

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

Step 4.1.2.1.2

Divide $y - 6$ by $1$ .

$y - 6 = \frac{0}{2}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $2$ .

$y - 6 = 0$

Step 4.2

Add $6$ to both sides of the equation.

$y = 6$