Solve 4/a-3+y/a+3-27/a^2-9

$$\frac { 4 } { a - 3 } + \frac { y } { a + 3 } - \frac { 27 } { a ^ { 2 } - 9 }$$

Evaluate

$$\frac{ay-3y+4a-15}{a^{2}-9}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-3 and a+3 is \left(a-3\right)\left(a+3\right). Multiply \frac{4}{a-3} times \frac{a+3}{a+3}. Multiply \frac{y}{a+3} times \frac{a-3}{a-3}.

$$\frac{4\left(a+3\right)}{\left(a-3\right)\left(a+3\right)}+\frac{y\left(a-3\right)}{\left(a-3\right)\left(a+3\right)}-\frac{27}{a^{2}-9}$$

Since \frac{4\left(a+3\right)}{\left(a-3\right)\left(a+3\right)} and \frac{y\left(a-3\right)}{\left(a-3\right)\left(a+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{4\left(a+3\right)+y\left(a-3\right)}{\left(a-3\right)\left(a+3\right)}-\frac{27}{a^{2}-9}$$

Do the multiplications in 4\left(a+3\right)+y\left(a-3\right).

$$\frac{4a+12+ya-3y}{\left(a-3\right)\left(a+3\right)}-\frac{27}{a^{2}-9}$$

Factor a^{2}-9.

$$\frac{4a+12+ya-3y}{\left(a-3\right)\left(a+3\right)}-\frac{27}{\left(a-3\right)\left(a+3\right)}$$

Since \frac{4a+12+ya-3y}{\left(a-3\right)\left(a+3\right)} and \frac{27}{\left(a-3\right)\left(a+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4a+12+ya-3y-27}{\left(a-3\right)\left(a+3\right)}$$

Combine like terms in 4a+12+ya-3y-27.

$$\frac{4a-15+ya-3y}{\left(a-3\right)\left(a+3\right)}$$

Expand \left(a-3\right)\left(a+3\right).

$$\frac{4a-15+ya-3y}{a^{2}-9}$$

Differentiate w.r.t. y

$$\frac{1}{a+3}$$