Why are the "extents of reaction" the same for all reactants and products?
Let's take a numerical example: $\ce{3 A + 1 B -&gt; 8 C + 2 D}$
Here : $a = -3, b = -1, c = 8, d = 2$.
Let's start from $100$ $mol$ of $\ce{A}$, $120$ $mol$ of $\ce{B}$, $1$ $mol$ $\ce{C}$, and no $\ce{D}$. Suppose that $10$ $mol$ $\ce{B}$ have reacted with $30$ $mol$ $\ce{A}$, and producing $80$ $mol$ $\ce{C}$ and $20$ $mol$ $\ce{D}$. At the end of the reaction, the remaining amounts are :
For $\ce{A}$ : $100$ $mol$ - $30$ $mol$ = $70$ $mol$ $\ce{A}$.
For $\ce{B}$ :  $\ce{120 $mol$ - $10$ $mol$ = $110$ $mol$}$ $\ce{B}$
For $\ce{C}$ : $\ce{1 $mol$ + 80 $mol$ = 81 $mol$}$ $\ce{C}$
$$\begin{align}
\xi_A(t) &amp;= \frac{n_A(t) - n_A(0)}{a} =\frac{70 - 100}{-3} = 10\tag{2.1}\\
\xi_B(t) &amp;= \frac{n_B(t) - n_B(0)}{b} =\frac{110-120}{-1}  = 10\tag{2.2}\\
\xi_C(t) &amp;= \frac{n_C(t) - n_C(0)}{c} =\frac{81 - 1}{8}    = 10\tag{2.3}\\
\xi_D(t) &amp;= \frac{n_D(t) - n_D(0)}{d} =\frac{20 - 0}{2}    = 10\tag{2.4}
\end{align}$$ The extent of reaction are the same for all products.