Algebra Examples

$2 y + \frac{9}{y + 3} - \frac{9}{3 - y}$

Step 1

To write $2 y$ as a fraction with a common denominator, multiply by $\frac{y + 3}{y + 3}$ .

$\frac{2 y (y + 3)}{y + 3} + \frac{9}{y + 3} - \frac{9}{3 - y}$

Step 2

Combine the numerators over the common denominator.

$\frac{2 y (y + 3) + 9}{y + 3} - \frac{9}{3 - y}$

Step 3

Simplify the numerator.

Tap for more steps...

Step 3.1

Apply the distributive property.

$\frac{2 y \cdot y + 2 y \cdot 3 + 9}{y + 3} - \frac{9}{3 - y}$

Step 3.2

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

Tap for more steps...

Step 3.2.1

Move $y$ .

$\frac{2 (y \cdot y) + 2 y \cdot 3 + 9}{y + 3} - \frac{9}{3 - y}$

Step 3.2.2

Multiply $y$ by $y$ .

$\frac{2 y^{2} + 2 y \cdot 3 + 9}{y + 3} - \frac{9}{3 - y}$

Step 3.3

Multiply $3$ by $2$ .

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

Step 4

To write $\frac{2 y^{2} + 6 y + 9}{y + 3}$ as a fraction with a common denominator, multiply by $\frac{3 - y}{3 - y}$ .

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

Step 5

To write $- \frac{9}{3 - y}$ as a fraction with a common denominator, multiply by $\frac{y + 3}{y + 3}$ .

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

Step 6

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

Tap for more steps...

Step 6.1

Multiply $\frac{2 y^{2} + 6 y + 9}{y + 3}$ by $\frac{3 - y}{3 - y}$ .

$\frac{(2 y^{2} + 6 y + 9) (3 - y)}{(y + 3) (3 - y)} - \frac{9}{3 - y} \cdot \frac{y + 3}{y + 3}$

Step 6.2

Multiply $\frac{9}{3 - y}$ by $\frac{y + 3}{y + 3}$ .

$\frac{(2 y^{2} + 6 y + 9) (3 - y)}{(y + 3) (3 - y)} - \frac{9 (y + 3)}{(3 - y) (y + 3)}$

Step 6.3

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

$\frac{(2 y^{2} + 6 y + 9) (3 - y)}{(3 - y) (y + 3)} - \frac{9 (y + 3)}{(3 - y) (y + 3)}$

Step 7

Combine the numerators over the common denominator.

$\frac{(2 y^{2} + 6 y + 9) (3 - y) - 9 (y + 3)}{(3 - y) (y + 3)}$

Step 8

Simplify the numerator.

Tap for more steps...

Step 8.1

Expand $(2 y^{2} + 6 y + 9) (3 - y)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{2 y^{2} \cdot 3 + 2 y^{2} (- y) + 6 y \cdot 3 + 6 y (- y) + 9 \cdot 3 + 9 (- y) - 9 (y + 3)}{(3 - y) (y + 3)}$

Step 8.2

Simplify each term.

Tap for more steps...

Step 8.2.1

Multiply $3$ by $2$ .

$\frac{6 y^{2} + 2 y^{2} (- y) + 6 y \cdot 3 + 6 y (- y) + 9 \cdot 3 + 9 (- y) - 9 (y + 3)}{(3 - y) (y + 3)}$

Step 8.2.2

Rewrite using the commutative property of multiplication.

$\frac{6 y^{2} + 2 \cdot - 1 y^{2} y + 6 y \cdot 3 + 6 y (- y) + 9 \cdot 3 + 9 (- y) - 9 (y + 3)}{(3 - y) (y + 3)}$

Step 8.2.3

Multiply $y^{2}$ by $y$ by adding the exponents.

Tap for more steps...

Step 8.2.3.1

Move $y$ .

$\frac{6 y^{2} + 2 \cdot - 1 (y \cdot y^{2}) + 6 y \cdot 3 + 6 y (- y) + 9 \cdot 3 + 9 (- y) - 9 (y + 3)}{(3 - y) (y + 3)}$

Step 8.2.3.2

Multiply $y$ by $y^{2}$ .

Tap for more steps...

Step 8.2.3.2.1

Raise $y$ to the power of $1$ .

$\frac{6 y^{2} + 2 \cdot - 1 (y^{1} y^{2}) + 6 y \cdot 3 + 6 y (- y) + 9 \cdot 3 + 9 (- y) - 9 (y + 3)}{(3 - y) (y + 3)}$

Step 8.2.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{6 y^{2} + 2 \cdot - 1 y^{1 + 2} + 6 y \cdot 3 + 6 y (- y) + 9 \cdot 3 + 9 (- y) - 9 (y + 3)}{(3 - y) (y + 3)}$

Step 8.2.3.3

Add $1$ and $2$ .

$\frac{6 y^{2} + 2 \cdot - 1 y^{3} + 6 y \cdot 3 + 6 y (- y) + 9 \cdot 3 + 9 (- y) - 9 (y + 3)}{(3 - y) (y + 3)}$

Step 8.2.4

Multiply $2$ by $- 1$ .

$\frac{6 y^{2} - 2 y^{3} + 6 y \cdot 3 + 6 y (- y) + 9 \cdot 3 + 9 (- y) - 9 (y + 3)}{(3 - y) (y + 3)}$

Step 8.2.5

Multiply $3$ by $6$ .

$\frac{6 y^{2} - 2 y^{3} + 18 y + 6 y (- y) + 9 \cdot 3 + 9 (- y) - 9 (y + 3)}{(3 - y) (y + 3)}$

Step 8.2.6

Rewrite using the commutative property of multiplication.

$\frac{6 y^{2} - 2 y^{3} + 18 y + 6 \cdot - 1 y \cdot y + 9 \cdot 3 + 9 (- y) - 9 (y + 3)}{(3 - y) (y + 3)}$

Step 8.2.7

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

Tap for more steps...

Step 8.2.7.1

Move $y$ .

$\frac{6 y^{2} - 2 y^{3} + 18 y + 6 \cdot - 1 (y \cdot y) + 9 \cdot 3 + 9 (- y) - 9 (y + 3)}{(3 - y) (y + 3)}$

Step 8.2.7.2

Multiply $y$ by $y$ .

$\frac{6 y^{2} - 2 y^{3} + 18 y + 6 \cdot - 1 y^{2} + 9 \cdot 3 + 9 (- y) - 9 (y + 3)}{(3 - y) (y + 3)}$

Step 8.2.8

Multiply $6$ by $- 1$ .

$\frac{6 y^{2} - 2 y^{3} + 18 y - 6 y^{2} + 9 \cdot 3 + 9 (- y) - 9 (y + 3)}{(3 - y) (y + 3)}$

Step 8.2.9

Multiply $9$ by $3$ .

$\frac{6 y^{2} - 2 y^{3} + 18 y - 6 y^{2} + 27 + 9 (- y) - 9 (y + 3)}{(3 - y) (y + 3)}$

Step 8.2.10

Multiply $- 1$ by $9$ .

$\frac{6 y^{2} - 2 y^{3} + 18 y - 6 y^{2} + 27 - 9 y - 9 (y + 3)}{(3 - y) (y + 3)}$

Step 8.3

Combine the opposite terms in $6 y^{2} - 2 y^{3} + 18 y - 6 y^{2} + 27 - 9 y$ .

Tap for more steps...

Step 8.3.1

Subtract $6 y^{2}$ from $6 y^{2}$ .

$\frac{- 2 y^{3} + 18 y + 0 + 27 - 9 y - 9 (y + 3)}{(3 - y) (y + 3)}$

Step 8.3.2

Add $- 2 y^{3} + 18 y$ and $0$ .

$\frac{- 2 y^{3} + 18 y + 27 - 9 y - 9 (y + 3)}{(3 - y) (y + 3)}$

Step 8.4

Subtract $9 y$ from $18 y$ .

$\frac{- 2 y^{3} + 9 y + 27 - 9 (y + 3)}{(3 - y) (y + 3)}$

Step 8.5

Apply the distributive property.

$\frac{- 2 y^{3} + 9 y + 27 - 9 y - 9 \cdot 3}{(3 - y) (y + 3)}$

Step 8.6

Multiply $- 9$ by $3$ .

$\frac{- 2 y^{3} + 9 y + 27 - 9 y - 27}{(3 - y) (y + 3)}$

Step 8.7

Subtract $9 y$ from $9 y$ .

$\frac{- 2 y^{3} + 0 + 27 - 27}{(3 - y) (y + 3)}$

Step 8.8

Add $- 2 y^{3}$ and $0$ .

$\frac{- 2 y^{3} + 27 - 27}{(3 - y) (y + 3)}$

Step 8.9

Subtract $27$ from $27$ .

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

Step 8.10

Add $- 2 y^{3}$ and $0$ .

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

Step 9

Move the negative in front of the fraction.

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