Add $ \dfrac{1}{a} $ and $ 5 $ to get $ \dfrac{ \color{purple}{ 5a+1 } }{ a }$.

Step 1: Write $ 5 $ 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 second fraction by $\color{blue}{a}$.

$$ \begin{aligned} \frac{1}{a} +5 & \xlongequal{\text{Step 1}} \frac{1}{a} + \frac{5}{\color{red}{1}} = \frac{ 1 }{ a } + \frac{ 5 \cdot \color{blue}{ a }}{ 1 \cdot \color{blue}{ a }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 1 } }{ a } + \frac{ \color{purple}{ 5a } }{ a }=\frac{ \color{purple}{ 5a+1 } }{ a } \end{aligned} $$

Multiply $2$ by $ \dfrac{a}{a} $ to get $ \dfrac{ 2a }{ a } $.

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.

$$ \begin{aligned} 2 \cdot \frac{a}{a} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} \cdot \frac{a}{a} \xlongequal{\text{Step 2}} \frac{ 2 \cdot a }{ 1 \cdot a } \xlongequal{\text{Step 3}} \frac{ 2a }{ a } \end{aligned} $$

Add $2$ and $ \dfrac{5a+1}{a} $ to get $ \dfrac{ \color{purple}{ 7a+1 } }{ a }$.

Step 1: Write $ 2 $ 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 $\color{blue}{ a }$.

$$ \begin{aligned} 2+ \frac{5a+1}{a} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} + \frac{5a+1}{a} = \frac{ 2 \cdot \color{blue}{ a }}{ 1 \cdot \color{blue}{ a }} + \frac{ 5a+1 }{ a } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2a } }{ a } + \frac{ \color{purple}{ 5a+1 } }{ a }=\frac{ \color{purple}{ 7a+1 } }{ a } \end{aligned} $$

Subtract $5$ from $ \dfrac{2a}{a} $ to get $ \dfrac{ \color{purple}{ -3a } }{ a }$.

Step 1: Write $ 5 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{a}$.

$$ \begin{aligned} \frac{2a}{a} -5 & \xlongequal{\text{Step 1}} \frac{2a}{a} - \frac{5}{\color{red}{1}} = \frac{ 2a }{ a } - \frac{ 5 \cdot \color{blue}{ a }}{ 1 \cdot \color{blue}{ a }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2a } }{ a } - \frac{ \color{purple}{ 5a } }{ a }=\frac{ \color{purple}{ -3a } }{ a } \end{aligned} $$

Subtract $ \dfrac{-3a}{a} $ from $ \dfrac{7a+1}{a} $ to get $ \dfrac{10a+1}{a} $.

To subtract expressions with the same denominators, we subtract the numerators and write the result over the common denominator.

$$ \begin{aligned} \frac{7a+1}{a} - \frac{-3a}{a} & = \frac{7a+1}{\color{blue}{a}} - \frac{-3a}{\color{blue}{a}} =\frac{ 7a+1 - \left( -3a \right) }{ \color{blue}{ a }} = \\[1ex] &= \frac{10a+1}{a} \end{aligned} $$