Solve 2y-6/y^2-9-y/y-1+y^2+2/y^2+2y-3

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

Evaluate

$$-\frac{y}{\left(y-1\right)\left(y+3\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{2y-6}{y^{2}-9}.

$$\frac{2\left(y-3\right)}{\left(y-3\right)\left(y+3\right)}-\frac{y}{y-1}+\frac{y^{2}+2}{y^{2}+2y-3}$$

Cancel out y-3 in both numerator and denominator.

$$\frac{2}{y+3}-\frac{y}{y-1}+\frac{y^{2}+2}{y^{2}+2y-3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y+3 and y-1 is \left(y-1\right)\left(y+3\right). Multiply \frac{2}{y+3} times \frac{y-1}{y-1}. Multiply \frac{y}{y-1} times \frac{y+3}{y+3}.

$$\frac{2\left(y-1\right)}{\left(y-1\right)\left(y+3\right)}-\frac{y\left(y+3\right)}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}$$

Since \frac{2\left(y-1\right)}{\left(y-1\right)\left(y+3\right)} and \frac{y\left(y+3\right)}{\left(y-1\right)\left(y+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(y-1\right)-y\left(y+3\right)}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}$$

Do the multiplications in 2\left(y-1\right)-y\left(y+3\right).

$$\frac{2y-2-y^{2}-3y}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}$$

Combine like terms in 2y-2-y^{2}-3y.

$$\frac{-y-2-y^{2}}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}$$

Factor y^{2}+2y-3.

$$\frac{-y-2-y^{2}}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{\left(y-1\right)\left(y+3\right)}$$

Since \frac{-y-2-y^{2}}{\left(y-1\right)\left(y+3\right)} and \frac{y^{2}+2}{\left(y-1\right)\left(y+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{-y-2-y^{2}+y^{2}+2}{\left(y-1\right)\left(y+3\right)}$$

Combine like terms in -y-2-y^{2}+y^{2}+2.

$$\frac{-y}{\left(y-1\right)\left(y+3\right)}$$

Expand \left(y-1\right)\left(y+3\right).

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

Expand

$$-\frac{y}{\left(y-1\right)\left(y+3\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{2y-6}{y^{2}-9}.

$$\frac{2\left(y-3\right)}{\left(y-3\right)\left(y+3\right)}-\frac{y}{y-1}+\frac{y^{2}+2}{y^{2}+2y-3}$$

Cancel out y-3 in both numerator and denominator.

$$\frac{2}{y+3}-\frac{y}{y-1}+\frac{y^{2}+2}{y^{2}+2y-3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y+3 and y-1 is \left(y-1\right)\left(y+3\right). Multiply \frac{2}{y+3} times \frac{y-1}{y-1}. Multiply \frac{y}{y-1} times \frac{y+3}{y+3}.

$$\frac{2\left(y-1\right)}{\left(y-1\right)\left(y+3\right)}-\frac{y\left(y+3\right)}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}$$

Since \frac{2\left(y-1\right)}{\left(y-1\right)\left(y+3\right)} and \frac{y\left(y+3\right)}{\left(y-1\right)\left(y+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(y-1\right)-y\left(y+3\right)}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}$$

Do the multiplications in 2\left(y-1\right)-y\left(y+3\right).

$$\frac{2y-2-y^{2}-3y}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}$$

Combine like terms in 2y-2-y^{2}-3y.

$$\frac{-y-2-y^{2}}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}$$

Factor y^{2}+2y-3.

$$\frac{-y-2-y^{2}}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{\left(y-1\right)\left(y+3\right)}$$

Since \frac{-y-2-y^{2}}{\left(y-1\right)\left(y+3\right)} and \frac{y^{2}+2}{\left(y-1\right)\left(y+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{-y-2-y^{2}+y^{2}+2}{\left(y-1\right)\left(y+3\right)}$$

Combine like terms in -y-2-y^{2}+y^{2}+2.

$$\frac{-y}{\left(y-1\right)\left(y+3\right)}$$

Expand \left(y-1\right)\left(y+3\right).

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