Algebra Examples

$\frac{5 y}{y + 10} + \frac{8 y^{2}}{y^{2} - 100} - \frac{9}{y - 10}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $100$ as $10^{2}$ .

$\frac{5 y}{y + 10} + \frac{8 y^{2}}{y^{2} - 10^{2}} - \frac{9}{y - 10}$

Step 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 $b = 10$ .

$\frac{5 y}{y + 10} + \frac{8 y^{2}}{(y + 10) (y - 10)} - \frac{9}{y - 10}$

Step 2

To write $\frac{5 y}{y + 10}$ as a fraction with a common denominator, multiply by $\frac{y - 10}{y - 10}$ .

$\frac{5 y}{y + 10} \cdot \frac{y - 10}{y - 10} + \frac{8 y^{2}}{(y + 10) (y - 10)} - \frac{9}{y - 10}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{5 y}{y + 10}$ by $\frac{y - 10}{y - 10}$ .

$\frac{5 y (y - 10)}{(y + 10) (y - 10)} + \frac{8 y^{2}}{(y + 10) (y - 10)} - \frac{9}{y - 10}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{5 y (y - 10) + 8 y^{2}}{(y + 10) (y - 10)} - \frac{9}{y - 10}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $y$ out of $5 y (y - 10) + 8 y^{2}$ .

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Step 4.1.1

Factor $y$ out of $5 y (y - 10)$ .

$\frac{y (5 (y - 10)) + 8 y^{2}}{(y + 10) (y - 10)} - \frac{9}{y - 10}$

Step 4.1.2

Factor $y$ out of $8 y^{2}$ .

$\frac{y (5 (y - 10)) + y (8 y)}{(y + 10) (y - 10)} - \frac{9}{y - 10}$

Step 4.1.3

Factor $y$ out of $y (5 (y - 10)) + y (8 y)$ .

$\frac{y (5 (y - 10) + 8 y)}{(y + 10) (y - 10)} - \frac{9}{y - 10}$

Step 4.2

Apply the distributive property.

$\frac{y (5 y + 5 \cdot - 10 + 8 y)}{(y + 10) (y - 10)} - \frac{9}{y - 10}$

Step 4.3

Multiply $5$ by $- 10$ .

$\frac{y (5 y - 50 + 8 y)}{(y + 10) (y - 10)} - \frac{9}{y - 10}$

Step 4.4

Add $5 y$ and $8 y$ .

$\frac{y (13 y - 50)}{(y + 10) (y - 10)} - \frac{9}{y - 10}$

Step 5

To write $- \frac{9}{y - 10}$ as a fraction with a common denominator, multiply by $\frac{y + 10}{y + 10}$ .

$\frac{y (13 y - 50)}{(y + 10) (y - 10)} - \frac{9}{y - 10} \cdot \frac{y + 10}{y + 10}$

Step 6

Write each expression with a common denominator of $(y + 10) (y - 10)$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1

Multiply $\frac{9}{y - 10}$ by $\frac{y + 10}{y + 10}$ .

$\frac{y (13 y - 50)}{(y + 10) (y - 10)} - \frac{9 (y + 10)}{(y - 10) (y + 10)}$

Step 6.2

Reorder the factors of $(y - 10) (y + 10)$ .

$\frac{y (13 y - 50)}{(y + 10) (y - 10)} - \frac{9 (y + 10)}{(y + 10) (y - 10)}$

Step 7

Combine the numerators over the common denominator.

$\frac{y (13 y - 50) - 9 (y + 10)}{(y + 10) (y - 10)}$

Step 8

Simplify the numerator.

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Step 8.1

Apply the distributive property.

$\frac{y (13 y) + y \cdot - 50 - 9 (y + 10)}{(y + 10) (y - 10)}$

Step 8.2

Rewrite using the commutative property of multiplication.

$\frac{13 y \cdot y + y \cdot - 50 - 9 (y + 10)}{(y + 10) (y - 10)}$

Step 8.3

Move $- 50$ to the left of $y$ .

$\frac{13 y \cdot y - 50 \cdot y - 9 (y + 10)}{(y + 10) (y - 10)}$

Step 8.4

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

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Step 8.4.1

Move $y$ .

$\frac{13 (y \cdot y) - 50 \cdot y - 9 (y + 10)}{(y + 10) (y - 10)}$

Step 8.4.2

Multiply $y$ by $y$ .

$\frac{13 y^{2} - 50 \cdot y - 9 (y + 10)}{(y + 10) (y - 10)}$

$\frac{13 y^{2} - 50 y - 9 (y + 10)}{(y + 10) (y - 10)}$

Step 8.5

Apply the distributive property.

$\frac{13 y^{2} - 50 y - 9 y - 9 \cdot 10}{(y + 10) (y - 10)}$

Step 8.6

Multiply $- 9$ by $10$ .

$\frac{13 y^{2} - 50 y - 9 y - 90}{(y + 10) (y - 10)}$

Step 8.7

Subtract $9 y$ from $- 50 y$ .

$\frac{13 y^{2} - 59 y - 90}{(y + 10) (y - 10)}$