Here is an example of a velocity (v) vs time (t) graph with constant acceleration (a):
The formula for a is:
$$ {\color{red} a} = \frac{{\color{blue} v_f} - {\color{blue} v_i}}{{\color{gray} t}} $$
Final velocity is represented as vf while initial velocity is represented as vi. Our first equation of motion is:
$$ \begin{gather} {\color{gray} t}{\color{red} a} = {\color{blue} v_f} - {\color{blue} v_i} \\ {\color{blue} v_f} = {\color{blue} v_i} + {\color{gray} t}{\color{red} a}\end{gather}$$
The average velocity (vavg) can be represented in two ways:
$$ \begin{gather} {\color{blue} v_{avg}} = \frac{{\color{blue} v_f} + {\color{blue} v_i}}{2} \\ {\color{blue} v_{avg}} = \frac{\color{green} s}{\color{gray} t}\end{gather}$$
If we equate them, we get our second equation:
$$ \begin{gather} \frac{{\color{blue} v_f} + {\color{blue} v_i}}{2} = \frac{\color{green} s}{\color{gray} t} \\ {\color{green} s} = \frac{({\color{blue} v_f} + {\color{blue} v_i}) {\color{gray} t}}{2 } \end{gather}$$
Suppose we substitute the first equation into the second one, we can get this:
$$ {\color{green} s} = \frac{(( {\color{blue} v_i} + {\color{gray} t}{\color{red} a}) + {\color{blue} v_i}) {\color{gray} t}}{2 } $$
Or this:
$${\color{green} s} = \frac{({\color{blue} v_f} + ( {\color{blue} v_f} - {\color{gray} t}{\color{red} a})) {\color{gray} t}}{2} $$
If we simplify, we get our third and fourth equation:
$$\begin{align} {\color{green} s} &= \frac{(( {\color{blue} v_i} + {\color{gray} t}{\color{red} a}) + {\color{blue} v_i}) {\color{gray} t}}{2} = \frac{ 2{\color{blue} v_i}{\color{gray} t} + {\color{gray} t^2}{\color{red} a}}{2} = {\color{blue} v_i}{\color{gray} t} + \frac{1}{2}{\color{gray} t^2}{\color{red} a} \\ {\color{green} s} &= \frac{({\color{blue} v_f} + ( {\color{blue} v_f} - {\color{gray} t}{\color{red} a})) {\color{gray} t}}{2} = \frac{2{\color{blue} v_f}{\color{gray} t} - {\color{gray} t^2}{\color{red} a}}{2} = {\color{blue} v_f}{\color{gray} t} - \frac{1}{2} {\color{gray} t^2}{\color{red} a} \end{align}$$
If we make t the subject in the first equation and s the subject in the second equation, then:
$$\begin{align} {\color{gray} t} &= \frac{({\color{blue} v_f} - {\color{blue} v_i})}{\color{red} a} \\ {\color{green} s} &= \frac{({\color{blue} v_f} + {\color{blue} v_i}){\color{gray} t}}{2} \end{align}$$
If we substitute t in the second equation, then:
$$ {\color{green} s} = \frac{({\color{blue} v_f} + {\color{blue} v_i})({\color{blue} v_f} - {\color{blue} v_i})}{2 {\color{red} a}} $$
If we simplify, we get our fifth equation:
$$\begin{align} 2 {\color{red} a} {\color{green} s} = {\color{blue} v_f^2} - {\color{blue} v_i^2} \\ {\color{blue} v_f^2} = {\color{blue} v_i^2} + 2 {\color{red} a} {\color{green} s} \end{align}$$