Solve y/y+4-y/y-4/frac{y{y+4}+y/y-4}

$$\frac { \frac { y } { y + 4 } - \frac { y } { y - 4 } } { \frac { y } { y + 4 } + \frac { y } { y - 4 } }$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y+4 and y-4 is \left(y-4\right)\left(y+4\right). Multiply \frac{y}{y+4} times \frac{y-4}{y-4}. Multiply \frac{y}{y-4} times \frac{y+4}{y+4}.

$$\frac{\frac{y\left(y-4\right)}{\left(y-4\right)\left(y+4\right)}-\frac{y\left(y+4\right)}{\left(y-4\right)\left(y+4\right)}}{\frac{y}{y+4}+\frac{y}{y-4}}$$

Since \frac{y\left(y-4\right)}{\left(y-4\right)\left(y+4\right)} and \frac{y\left(y+4\right)}{\left(y-4\right)\left(y+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{y\left(y-4\right)-y\left(y+4\right)}{\left(y-4\right)\left(y+4\right)}}{\frac{y}{y+4}+\frac{y}{y-4}}$$

Do the multiplications in y\left(y-4\right)-y\left(y+4\right).

$$\frac{\frac{y^{2}-4y-y^{2}-4y}{\left(y-4\right)\left(y+4\right)}}{\frac{y}{y+4}+\frac{y}{y-4}}$$

Combine like terms in y^{2}-4y-y^{2}-4y.

$$\frac{\frac{-8y}{\left(y-4\right)\left(y+4\right)}}{\frac{y}{y+4}+\frac{y}{y-4}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y+4 and y-4 is \left(y-4\right)\left(y+4\right). Multiply \frac{y}{y+4} times \frac{y-4}{y-4}. Multiply \frac{y}{y-4} times \frac{y+4}{y+4}.

$$\frac{\frac{-8y}{\left(y-4\right)\left(y+4\right)}}{\frac{y\left(y-4\right)}{\left(y-4\right)\left(y+4\right)}+\frac{y\left(y+4\right)}{\left(y-4\right)\left(y+4\right)}}$$

Since \frac{y\left(y-4\right)}{\left(y-4\right)\left(y+4\right)} and \frac{y\left(y+4\right)}{\left(y-4\right)\left(y+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-8y}{\left(y-4\right)\left(y+4\right)}}{\frac{y\left(y-4\right)+y\left(y+4\right)}{\left(y-4\right)\left(y+4\right)}}$$

Do the multiplications in y\left(y-4\right)+y\left(y+4\right).

$$\frac{\frac{-8y}{\left(y-4\right)\left(y+4\right)}}{\frac{y^{2}-4y+y^{2}+4y}{\left(y-4\right)\left(y+4\right)}}$$

Combine like terms in y^{2}-4y+y^{2}+4y.

$$\frac{\frac{-8y}{\left(y-4\right)\left(y+4\right)}}{\frac{2y^{2}}{\left(y-4\right)\left(y+4\right)}}$$

Divide \frac{-8y}{\left(y-4\right)\left(y+4\right)} by \frac{2y^{2}}{\left(y-4\right)\left(y+4\right)} by multiplying \frac{-8y}{\left(y-4\right)\left(y+4\right)} by the reciprocal of \frac{2y^{2}}{\left(y-4\right)\left(y+4\right)}.

$$\frac{-8y\left(y-4\right)\left(y+4\right)}{\left(y-4\right)\left(y+4\right)\times 2y^{2}}$$

Cancel out 2y\left(y-4\right)\left(y+4\right) in both numerator and denominator.

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y+4 and y-4 is \left(y-4\right)\left(y+4\right). Multiply \frac{y}{y+4} times \frac{y-4}{y-4}. Multiply \frac{y}{y-4} times \frac{y+4}{y+4}.

$$\frac{\frac{y\left(y-4\right)}{\left(y-4\right)\left(y+4\right)}-\frac{y\left(y+4\right)}{\left(y-4\right)\left(y+4\right)}}{\frac{y}{y+4}+\frac{y}{y-4}}$$

Since \frac{y\left(y-4\right)}{\left(y-4\right)\left(y+4\right)} and \frac{y\left(y+4\right)}{\left(y-4\right)\left(y+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{y\left(y-4\right)-y\left(y+4\right)}{\left(y-4\right)\left(y+4\right)}}{\frac{y}{y+4}+\frac{y}{y-4}}$$

Do the multiplications in y\left(y-4\right)-y\left(y+4\right).

$$\frac{\frac{y^{2}-4y-y^{2}-4y}{\left(y-4\right)\left(y+4\right)}}{\frac{y}{y+4}+\frac{y}{y-4}}$$

Combine like terms in y^{2}-4y-y^{2}-4y.

$$\frac{\frac{-8y}{\left(y-4\right)\left(y+4\right)}}{\frac{y}{y+4}+\frac{y}{y-4}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y+4 and y-4 is \left(y-4\right)\left(y+4\right). Multiply \frac{y}{y+4} times \frac{y-4}{y-4}. Multiply \frac{y}{y-4} times \frac{y+4}{y+4}.

$$\frac{\frac{-8y}{\left(y-4\right)\left(y+4\right)}}{\frac{y\left(y-4\right)}{\left(y-4\right)\left(y+4\right)}+\frac{y\left(y+4\right)}{\left(y-4\right)\left(y+4\right)}}$$

Since \frac{y\left(y-4\right)}{\left(y-4\right)\left(y+4\right)} and \frac{y\left(y+4\right)}{\left(y-4\right)\left(y+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-8y}{\left(y-4\right)\left(y+4\right)}}{\frac{y\left(y-4\right)+y\left(y+4\right)}{\left(y-4\right)\left(y+4\right)}}$$

Do the multiplications in y\left(y-4\right)+y\left(y+4\right).

$$\frac{\frac{-8y}{\left(y-4\right)\left(y+4\right)}}{\frac{y^{2}-4y+y^{2}+4y}{\left(y-4\right)\left(y+4\right)}}$$

Combine like terms in y^{2}-4y+y^{2}+4y.

$$\frac{\frac{-8y}{\left(y-4\right)\left(y+4\right)}}{\frac{2y^{2}}{\left(y-4\right)\left(y+4\right)}}$$

Divide \frac{-8y}{\left(y-4\right)\left(y+4\right)} by \frac{2y^{2}}{\left(y-4\right)\left(y+4\right)} by multiplying \frac{-8y}{\left(y-4\right)\left(y+4\right)} by the reciprocal of \frac{2y^{2}}{\left(y-4\right)\left(y+4\right)}.

$$\frac{-8y\left(y-4\right)\left(y+4\right)}{\left(y-4\right)\left(y+4\right)\times 2y^{2}}$$

Cancel out 2y\left(y-4\right)\left(y+4\right) in both numerator and denominator.

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