Add $ \dfrac{1}{3y} $ and $ \dfrac{1}{x-7} $ to get $ \dfrac{ \color{purple}{ x+3y-7 } }{ 3xy-21y }$.

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}{ x-7 }$ and the second by $\color{blue}{ 3y }$.

$$ \begin{aligned} \frac{1}{3y} + \frac{1}{x-7} & = \frac{ 1 \cdot \color{blue}{ \left( x-7 \right) }}{ 3y \cdot \color{blue}{ \left( x-7 \right) }} + \frac{ 1 \cdot \color{blue}{ 3y }}{ \left( x-7 \right) \cdot \color{blue}{ 3y }} = \\[1ex] &=\frac{ \color{purple}{ x-7 } }{ 3xy-21y } + \frac{ \color{purple}{ 3y } }{ 3xy-21y }=\frac{ \color{purple}{ x+3y-7 } }{ 3xy-21y } \end{aligned} $$

Multiply $3$ by $ \dfrac{y}{5x^2-35x} $ to get $ \dfrac{ 3y }{ 5x^2-35x } $.

Step 1: Write $ 3 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 3 \cdot \frac{y}{5x^2-35x} & \xlongequal{\text{Step 1}} \frac{3}{\color{red}{1}} \cdot \frac{y}{5x^2-35x} \xlongequal{\text{Step 2}} \frac{ 3 \cdot y }{ 1 \cdot \left( 5x^2-35x \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 3y }{ 5x^2-35x } \end{aligned} $$

Add $ \dfrac{1}{3y} $ and $ \dfrac{1}{x-7} $ to get $ \dfrac{ \color{purple}{ x+3y-7 } }{ 3xy-21y }$.

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}{ x-7 }$ and the second by $\color{blue}{ 3y }$.

$$ \begin{aligned} \frac{1}{3y} + \frac{1}{x-7} & = \frac{ 1 \cdot \color{blue}{ \left( x-7 \right) }}{ 3y \cdot \color{blue}{ \left( x-7 \right) }} + \frac{ 1 \cdot \color{blue}{ 3y }}{ \left( x-7 \right) \cdot \color{blue}{ 3y }} = \\[1ex] &=\frac{ \color{purple}{ x-7 } }{ 3xy-21y } + \frac{ \color{purple}{ 3y } }{ 3xy-21y }=\frac{ \color{purple}{ x+3y-7 } }{ 3xy-21y } \end{aligned} $$

Add $ \dfrac{1}{5x} $ and $ \dfrac{3y}{5x^2-35x} $ to get $ \dfrac{ \color{purple}{ 5x^2+15xy-35x } }{ 25x^3-175x^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}{ 5x^2-35x }$ and the second by $\color{blue}{ 5x }$.

$$ \begin{aligned} \frac{1}{5x} + \frac{3y}{5x^2-35x} & = \frac{ 1 \cdot \color{blue}{ \left( 5x^2-35x \right) }}{ 5x \cdot \color{blue}{ \left( 5x^2-35x \right) }} + \frac{ 3y \cdot \color{blue}{ 5x }}{ \left( 5x^2-35x \right) \cdot \color{blue}{ 5x }} = \\[1ex] &=\frac{ \color{purple}{ 5x^2-35x } }{ 25x^3-175x^2 } + \frac{ \color{purple}{ 15xy } }{ 25x^3-175x^2 } = \\[1ex] &=\frac{ \color{purple}{ 5x^2+15xy-35x } }{ 25x^3-175x^2 } \end{aligned} $$

Divide $ \dfrac{x+3y-7}{3xy-21y} $ by $ \dfrac{x+3y-7}{5x^2-35x} $ to get $ \dfrac{ 5x^2-35x }{ 3xy-21y } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Cancel $ \color{blue}{ x+3y-7 } $ in first and second fraction.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{x+3y-7}{3xy-21y} }{ \frac{\color{blue}{x+3y-7}}{\color{blue}{5x^2-35x}} } & \xlongequal{\text{Step 1}} \frac{x+3y-7}{3xy-21y} \cdot \frac{\color{blue}{5x^2-35x}}{\color{blue}{x+3y-7}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{\color{blue}{1}}{3xy-21y} \cdot \frac{5x^2-35x}{\color{blue}{1}} \xlongequal{\text{Step 3}} \frac{ 1 \cdot \left( 5x^2-35x \right) }{ \left( 3xy-21y \right) \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ 5x^2-35x }{ 3xy-21y } \end{aligned} $$