Add $z$ and $ \dfrac{z}{10z-9} $ to get $ \dfrac{ \color{purple}{ 10z^2-8z } }{ 10z-9 }$.

Step 1: Write $ z $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 10z-9 }$.

$$ \begin{aligned} z+ \frac{z}{10z-9} & \xlongequal{\text{Step 1}} \frac{z}{\color{red}{1}} + \frac{z}{10z-9} = \frac{ z \cdot \color{blue}{ \left( 10z-9 \right) }}{ 1 \cdot \color{blue}{ \left( 10z-9 \right) }} + \frac{ z }{ 10z-9 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 10z^2-9z } }{ 10z-9 } + \frac{ \color{purple}{ z } }{ 10z-9 }=\frac{ \color{purple}{ 10z^2-8z } }{ 10z-9 } \end{aligned} $$