Solve y+7/y^2+3y-28-7/y^2-49

$$\frac { y + 7 } { y ^ { 2 } + 3 y - 28 } - \frac { 7 } { y ^ { 2 } - 49 }$$

Evaluate

$$\frac{y^{2}-7y-21}{\left(y-4\right)\left(y^{2}-49\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{y+7}{y^{2}+3y-28}.

$$\frac{y+7}{\left(y-4\right)\left(y+7\right)}-\frac{7}{y^{2}-49}$$

Cancel out y+7 in both numerator and denominator.

$$\frac{1}{y-4}-\frac{7}{y^{2}-49}$$

Factor y^{2}-49.

$$\frac{1}{y-4}-\frac{7}{\left(y-7\right)\left(y+7\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y-4 and \left(y-7\right)\left(y+7\right) is \left(y-7\right)\left(y-4\right)\left(y+7\right). Multiply \frac{1}{y-4} times \frac{\left(y-7\right)\left(y+7\right)}{\left(y-7\right)\left(y+7\right)}. Multiply \frac{7}{\left(y-7\right)\left(y+7\right)} times \frac{y-4}{y-4}.

$$\frac{\left(y-7\right)\left(y+7\right)}{\left(y-7\right)\left(y-4\right)\left(y+7\right)}-\frac{7\left(y-4\right)}{\left(y-7\right)\left(y-4\right)\left(y+7\right)}$$

Since \frac{\left(y-7\right)\left(y+7\right)}{\left(y-7\right)\left(y-4\right)\left(y+7\right)} and \frac{7\left(y-4\right)}{\left(y-7\right)\left(y-4\right)\left(y+7\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(y-7\right)\left(y+7\right)-7\left(y-4\right)}{\left(y-7\right)\left(y-4\right)\left(y+7\right)}$$

Do the multiplications in \left(y-7\right)\left(y+7\right)-7\left(y-4\right).

$$\frac{y^{2}+7y-7y-49-7y+28}{\left(y-7\right)\left(y-4\right)\left(y+7\right)}$$

Combine like terms in y^{2}+7y-7y-49-7y+28.

$$\frac{y^{2}-7y-21}{\left(y-7\right)\left(y-4\right)\left(y+7\right)}$$

Expand \left(y-7\right)\left(y-4\right)\left(y+7\right).

$$\frac{y^{2}-7y-21}{y^{3}-4y^{2}-49y+196}$$

Expand

$$\frac{y^{2}-7y-21}{\left(y-4\right)\left(y^{2}-49\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{y+7}{y^{2}+3y-28}.

$$\frac{y+7}{\left(y-4\right)\left(y+7\right)}-\frac{7}{y^{2}-49}$$

Cancel out y+7 in both numerator and denominator.

$$\frac{1}{y-4}-\frac{7}{y^{2}-49}$$

Factor y^{2}-49.

$$\frac{1}{y-4}-\frac{7}{\left(y-7\right)\left(y+7\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y-4 and \left(y-7\right)\left(y+7\right) is \left(y-7\right)\left(y-4\right)\left(y+7\right). Multiply \frac{1}{y-4} times \frac{\left(y-7\right)\left(y+7\right)}{\left(y-7\right)\left(y+7\right)}. Multiply \frac{7}{\left(y-7\right)\left(y+7\right)} times \frac{y-4}{y-4}.

$$\frac{\left(y-7\right)\left(y+7\right)}{\left(y-7\right)\left(y-4\right)\left(y+7\right)}-\frac{7\left(y-4\right)}{\left(y-7\right)\left(y-4\right)\left(y+7\right)}$$

Since \frac{\left(y-7\right)\left(y+7\right)}{\left(y-7\right)\left(y-4\right)\left(y+7\right)} and \frac{7\left(y-4\right)}{\left(y-7\right)\left(y-4\right)\left(y+7\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(y-7\right)\left(y+7\right)-7\left(y-4\right)}{\left(y-7\right)\left(y-4\right)\left(y+7\right)}$$

Do the multiplications in \left(y-7\right)\left(y+7\right)-7\left(y-4\right).

$$\frac{y^{2}+7y-7y-49-7y+28}{\left(y-7\right)\left(y-4\right)\left(y+7\right)}$$

Combine like terms in y^{2}+7y-7y-49-7y+28.

$$\frac{y^{2}-7y-21}{\left(y-7\right)\left(y-4\right)\left(y+7\right)}$$

Expand \left(y-7\right)\left(y-4\right)\left(y+7\right).

$$\frac{y^{2}-7y-21}{y^{3}-4y^{2}-49y+196}$$