Solve -2y/y^2-1-1/y+1/frac{y{y-1}}

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

Evaluate

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

Short Solution Steps

Factor y^{2}-1.

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

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

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

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

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

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

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

Combine like terms in -2y-y+1.

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

Divide \frac{-3y+1}{\left(y-1\right)\left(y+1\right)} by \frac{y}{y-1} by multiplying \frac{-3y+1}{\left(y-1\right)\left(y+1\right)} by the reciprocal of \frac{y}{y-1}.

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

Cancel out y-1 in both numerator and denominator.

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

Use the distributive property to multiply y by y+1.

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

Expand

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

Short Solution Steps

Factor y^{2}-1.

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

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

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

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

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

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

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

Combine like terms in -2y-y+1.

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

Divide \frac{-3y+1}{\left(y-1\right)\left(y+1\right)} by \frac{y}{y-1} by multiplying \frac{-3y+1}{\left(y-1\right)\left(y+1\right)} by the reciprocal of \frac{y}{y-1}.

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

Cancel out y-1 in both numerator and denominator.

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

Use the distributive property to multiply y by y+1.

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