Solve y/y+7-y/y-7/frac{y{y+7}+y/y-7}

$$\frac { \frac { y } { y + 7 } - \frac { y } { y - 7 } } { \frac { y } { y + 7 } + \frac { y } { y - 7 } }$$

Evaluate

$$-\frac{7}{y}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y+7 and y-7 is \left(y-7\right)\left(y+7\right). Multiply \frac{y}{y+7} times \frac{y-7}{y-7}. Multiply \frac{y}{y-7} times \frac{y+7}{y+7}.

$$\frac{\frac{y\left(y-7\right)}{\left(y-7\right)\left(y+7\right)}-\frac{y\left(y+7\right)}{\left(y-7\right)\left(y+7\right)}}{\frac{y}{y+7}+\frac{y}{y-7}}$$

Since \frac{y\left(y-7\right)}{\left(y-7\right)\left(y+7\right)} and \frac{y\left(y+7\right)}{\left(y-7\right)\left(y+7\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{y\left(y-7\right)-y\left(y+7\right)}{\left(y-7\right)\left(y+7\right)}}{\frac{y}{y+7}+\frac{y}{y-7}}$$

Do the multiplications in y\left(y-7\right)-y\left(y+7\right).

$$\frac{\frac{y^{2}-7y-y^{2}-7y}{\left(y-7\right)\left(y+7\right)}}{\frac{y}{y+7}+\frac{y}{y-7}}$$

Combine like terms in y^{2}-7y-y^{2}-7y.

$$\frac{\frac{-14y}{\left(y-7\right)\left(y+7\right)}}{\frac{y}{y+7}+\frac{y}{y-7}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y+7 and y-7 is \left(y-7\right)\left(y+7\right). Multiply \frac{y}{y+7} times \frac{y-7}{y-7}. Multiply \frac{y}{y-7} times \frac{y+7}{y+7}.

$$\frac{\frac{-14y}{\left(y-7\right)\left(y+7\right)}}{\frac{y\left(y-7\right)}{\left(y-7\right)\left(y+7\right)}+\frac{y\left(y+7\right)}{\left(y-7\right)\left(y+7\right)}}$$

Since \frac{y\left(y-7\right)}{\left(y-7\right)\left(y+7\right)} and \frac{y\left(y+7\right)}{\left(y-7\right)\left(y+7\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-14y}{\left(y-7\right)\left(y+7\right)}}{\frac{y\left(y-7\right)+y\left(y+7\right)}{\left(y-7\right)\left(y+7\right)}}$$

Do the multiplications in y\left(y-7\right)+y\left(y+7\right).

$$\frac{\frac{-14y}{\left(y-7\right)\left(y+7\right)}}{\frac{y^{2}-7y+y^{2}+7y}{\left(y-7\right)\left(y+7\right)}}$$

Combine like terms in y^{2}-7y+y^{2}+7y.

$$\frac{\frac{-14y}{\left(y-7\right)\left(y+7\right)}}{\frac{2y^{2}}{\left(y-7\right)\left(y+7\right)}}$$

Divide \frac{-14y}{\left(y-7\right)\left(y+7\right)} by \frac{2y^{2}}{\left(y-7\right)\left(y+7\right)} by multiplying \frac{-14y}{\left(y-7\right)\left(y+7\right)} by the reciprocal of \frac{2y^{2}}{\left(y-7\right)\left(y+7\right)}.

$$\frac{-14y\left(y-7\right)\left(y+7\right)}{\left(y-7\right)\left(y+7\right)\times 2y^{2}}$$

Cancel out 2y\left(y-7\right)\left(y+7\right) in both numerator and denominator.

$$\frac{-7}{y}$$

Expand

$$-\frac{7}{y}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y+7 and y-7 is \left(y-7\right)\left(y+7\right). Multiply \frac{y}{y+7} times \frac{y-7}{y-7}. Multiply \frac{y}{y-7} times \frac{y+7}{y+7}.

$$\frac{\frac{y\left(y-7\right)}{\left(y-7\right)\left(y+7\right)}-\frac{y\left(y+7\right)}{\left(y-7\right)\left(y+7\right)}}{\frac{y}{y+7}+\frac{y}{y-7}}$$

Since \frac{y\left(y-7\right)}{\left(y-7\right)\left(y+7\right)} and \frac{y\left(y+7\right)}{\left(y-7\right)\left(y+7\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{y\left(y-7\right)-y\left(y+7\right)}{\left(y-7\right)\left(y+7\right)}}{\frac{y}{y+7}+\frac{y}{y-7}}$$

Do the multiplications in y\left(y-7\right)-y\left(y+7\right).

$$\frac{\frac{y^{2}-7y-y^{2}-7y}{\left(y-7\right)\left(y+7\right)}}{\frac{y}{y+7}+\frac{y}{y-7}}$$

Combine like terms in y^{2}-7y-y^{2}-7y.

$$\frac{\frac{-14y}{\left(y-7\right)\left(y+7\right)}}{\frac{y}{y+7}+\frac{y}{y-7}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y+7 and y-7 is \left(y-7\right)\left(y+7\right). Multiply \frac{y}{y+7} times \frac{y-7}{y-7}. Multiply \frac{y}{y-7} times \frac{y+7}{y+7}.

$$\frac{\frac{-14y}{\left(y-7\right)\left(y+7\right)}}{\frac{y\left(y-7\right)}{\left(y-7\right)\left(y+7\right)}+\frac{y\left(y+7\right)}{\left(y-7\right)\left(y+7\right)}}$$

Since \frac{y\left(y-7\right)}{\left(y-7\right)\left(y+7\right)} and \frac{y\left(y+7\right)}{\left(y-7\right)\left(y+7\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-14y}{\left(y-7\right)\left(y+7\right)}}{\frac{y\left(y-7\right)+y\left(y+7\right)}{\left(y-7\right)\left(y+7\right)}}$$

Do the multiplications in y\left(y-7\right)+y\left(y+7\right).

$$\frac{\frac{-14y}{\left(y-7\right)\left(y+7\right)}}{\frac{y^{2}-7y+y^{2}+7y}{\left(y-7\right)\left(y+7\right)}}$$

Combine like terms in y^{2}-7y+y^{2}+7y.

$$\frac{\frac{-14y}{\left(y-7\right)\left(y+7\right)}}{\frac{2y^{2}}{\left(y-7\right)\left(y+7\right)}}$$

Divide \frac{-14y}{\left(y-7\right)\left(y+7\right)} by \frac{2y^{2}}{\left(y-7\right)\left(y+7\right)} by multiplying \frac{-14y}{\left(y-7\right)\left(y+7\right)} by the reciprocal of \frac{2y^{2}}{\left(y-7\right)\left(y+7\right)}.

$$\frac{-14y\left(y-7\right)\left(y+7\right)}{\left(y-7\right)\left(y+7\right)\times 2y^{2}}$$

Cancel out 2y\left(y-7\right)\left(y+7\right) in both numerator and denominator.

$$\frac{-7}{y}$$