Multiply $ \dfrac{2}{3} $ by $ x $ to get $ \dfrac{ 2x }{ 3 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $ \dfrac{3}{4} $ by $ y $ to get $ \dfrac{ 3y }{ 4 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{3}{4} \cdot y & \xlongequal{\text{Step 1}} \frac{3}{4} \cdot \frac{y}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 3 \cdot y }{ 4 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 3y }{ 4 } \end{aligned} $$

Multiply $ \dfrac{1}{3} $ by $ y $ to get $ \dfrac{ y }{ 3 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{2x}{3} $ and $ y $ to get $ \dfrac{ \color{purple}{ 2x+3y } }{ 3 }$.

Step 1: Write $ y $ 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 $\color{blue}{3}$.

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

Multiply $ \dfrac{3}{4} $ by $ y $ to get $ \dfrac{ 3y }{ 4 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{3}{4} \cdot y & \xlongequal{\text{Step 1}} \frac{3}{4} \cdot \frac{y}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 3 \cdot y }{ 4 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 3y }{ 4 } \end{aligned} $$

Multiply $ \dfrac{1}{3} $ by $ y $ to get $ \dfrac{ y }{ 3 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Subtract $ \dfrac{3y}{4} $ from $ \dfrac{2x+3y}{3} $ to get $ \dfrac{ \color{purple}{ 8x+3y } }{ 12 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ 4 }$ and the second by $\color{blue}{ 3 }$.

$$ \begin{aligned} \frac{2x+3y}{3} - \frac{3y}{4} & = \frac{ \left( 2x+3y \right) \cdot \color{blue}{ 4 }}{ 3 \cdot \color{blue}{ 4 }} - \frac{ 3y \cdot \color{blue}{ 3 }}{ 4 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ 8x+12y } }{ 12 } - \frac{ \color{purple}{ 9y } }{ 12 }=\frac{ \color{purple}{ 8x+3y } }{ 12 } \end{aligned} $$

Multiply $ \dfrac{1}{3} $ by $ y $ to get $ \dfrac{ y }{ 3 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Subtract $ \dfrac{y}{3} $ from $ \dfrac{8x+3y}{12} $ to get $ \dfrac{ \color{purple}{ 8x-y } }{ 12 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{4}$.

$$ \begin{aligned} \frac{8x+3y}{12} - \frac{y}{3} & = \frac{ 8x+3y }{ 12 } - \frac{ y \cdot \color{blue}{ 4 }}{ 3 \cdot \color{blue}{ 4 }} = \\[1ex] &=\frac{ \color{purple}{ 8x+3y } }{ 12 } - \frac{ \color{purple}{ 4y } }{ 12 }=\frac{ \color{purple}{ 8x-y } }{ 12 } \end{aligned} $$