Solve 9y+3/y^2-11y+18+y+3/y-9

$$\frac { 9 y + 3 } { y ^ { 2 } - 11 y + 18 } + \frac { y + 3 } { y - 9 }$$

Evaluate

$$\frac{y^{2}+10y-3}{\left(y-9\right)\left(y-2\right)}$$

Short Solution Steps

Factor y^{2}-11y+18.

$$\frac{9y+3}{\left(y-9\right)\left(y-2\right)}+\frac{y+3}{y-9}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(y-9\right)\left(y-2\right) and y-9 is \left(y-9\right)\left(y-2\right). Multiply \frac{y+3}{y-9} times \frac{y-2}{y-2}.

$$\frac{9y+3}{\left(y-9\right)\left(y-2\right)}+\frac{\left(y+3\right)\left(y-2\right)}{\left(y-9\right)\left(y-2\right)}$$

Since \frac{9y+3}{\left(y-9\right)\left(y-2\right)} and \frac{\left(y+3\right)\left(y-2\right)}{\left(y-9\right)\left(y-2\right)} have the same denominator, add them by adding their numerators.

$$\frac{9y+3+\left(y+3\right)\left(y-2\right)}{\left(y-9\right)\left(y-2\right)}$$

Do the multiplications in 9y+3+\left(y+3\right)\left(y-2\right).

$$\frac{9y+3+y^{2}-2y+3y-6}{\left(y-9\right)\left(y-2\right)}$$

Combine like terms in 9y+3+y^{2}-2y+3y-6.

$$\frac{10y-3+y^{2}}{\left(y-9\right)\left(y-2\right)}$$

Expand \left(y-9\right)\left(y-2\right).

$$\frac{10y-3+y^{2}}{y^{2}-11y+18}$$

Expand

$$\frac{y^{2}+10y-3}{\left(y-9\right)\left(y-2\right)}$$

Short Solution Steps

Factor y^{2}-11y+18.

$$\frac{9y+3}{\left(y-9\right)\left(y-2\right)}+\frac{y+3}{y-9}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(y-9\right)\left(y-2\right) and y-9 is \left(y-9\right)\left(y-2\right). Multiply \frac{y+3}{y-9} times \frac{y-2}{y-2}.

$$\frac{9y+3}{\left(y-9\right)\left(y-2\right)}+\frac{\left(y+3\right)\left(y-2\right)}{\left(y-9\right)\left(y-2\right)}$$

Since \frac{9y+3}{\left(y-9\right)\left(y-2\right)} and \frac{\left(y+3\right)\left(y-2\right)}{\left(y-9\right)\left(y-2\right)} have the same denominator, add them by adding their numerators.

$$\frac{9y+3+\left(y+3\right)\left(y-2\right)}{\left(y-9\right)\left(y-2\right)}$$

Do the multiplications in 9y+3+\left(y+3\right)\left(y-2\right).

$$\frac{9y+3+y^{2}-2y+3y-6}{\left(y-9\right)\left(y-2\right)}$$

Combine like terms in 9y+3+y^{2}-2y+3y-6.

$$\frac{10y-3+y^{2}}{\left(y-9\right)\left(y-2\right)}$$

Expand \left(y-9\right)\left(y-2\right).

$$\frac{10y-3+y^{2}}{y^{2}-11y+18}$$