Subtract $ \dfrac{1}{a} $ from $ x $ to get $ \dfrac{ \color{purple}{ ax-1 } }{ a }$.

Step 1: Write $ x $ 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 first fraction by $\color{blue}{ a }$.

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

Multiply $ \dfrac{ax-1}{a} $ by $ x-y $ to get $ \dfrac{ ax^2-axy-x+y }{ a } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $ \dfrac{ax^2-axy-x+y}{a} $ by $ x-z $ to get $ \dfrac{ax^3-ax^2y-ax^2z+axyz-x^2+xy+xz-yz}{a} $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ax^2-axy-x+y}{a} \cdot x-z & \xlongequal{\text{Step 1}} \frac{ax^2-axy-x+y}{a} \cdot \frac{x-z}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( ax^2-axy-x+y \right) \cdot \left( x-z \right) }{ a \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ ax^3-ax^2z-ax^2y+axyz-x^2+xz+xy-yz }{ a } = \frac{ax^3-ax^2y-ax^2z+axyz-x^2+xy+xz-yz}{a} \end{aligned} $$