Simplify $ \dfrac{y^2-2y}{y^2+7y-18} $ to $ \dfrac{y}{y+9} $.

Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{y-2}$.

$$ \begin{aligned} \frac{y^2-2y}{y^2+7y-18} & =\frac{ y \cdot \color{blue}{ \left( y-2 \right) }}{ \left( y+9 \right) \cdot \color{blue}{ \left( y-2 \right) }} = \\[1ex] &= \frac{y}{y+9} \end{aligned} $$

Simplify $ \dfrac{y^2-81}{y^2-11y+18} $ to $ \dfrac{y+9}{y-2} $.

Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{y-9}$.

$$ \begin{aligned} \frac{y^2-81}{y^2-11y+18} & =\frac{ \left( y+9 \right) \cdot \color{blue}{ \left( y-9 \right) }}{ \left( y-2 \right) \cdot \color{blue}{ \left( y-9 \right) }} = \\[1ex] &= \frac{y+9}{y-2} \end{aligned} $$

Multiply $ \dfrac{y}{y+9} $ by $ \dfrac{y+9}{y-2} $ to get $ \dfrac{ y }{ y-2 } $.

Step 1: Cancel $ \color{red}{ y+9 } $ in first and second fraction.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{y}{y+9} \cdot \frac{y+9}{y-2} & \xlongequal{\text{Step 1}} \frac{y}{\color{red}{1}} \cdot \frac{\color{red}{1}}{y-2} \xlongequal{\text{Step 2}} \frac{ y \cdot 1 }{ 1 \cdot \left( y-2 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ y }{ y-2 } \end{aligned} $$