Multiply $2$ by $ \dfrac{z}{3z-1} $ to get $ \dfrac{ 2z }{ 3z-1 } $.

Step 1: Write $ 2 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{2z}{3z-1} $ and $ \dfrac{z}{1-3z} $ to get $ \dfrac{ \color{purple}{ -z } }{ -3z+1 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ -1 }$.

$$ \begin{aligned} \frac{2z}{3z-1} + \frac{z}{1-3z} & = \frac{ 2z \cdot \color{blue}{ \left( -1 \right) }}{ \left( 3z-1 \right) \cdot \color{blue}{ \left( -1 \right) }} + \frac{ z }{ 1-3z } = \\[1ex] &=\frac{ \color{purple}{ -2z } }{ -3z+1 } + \frac{ \color{purple}{ z } }{ -3z+1 }=\frac{ \color{purple}{ -z } }{ -3z+1 } \end{aligned} $$