An indeterminate  is simply a variable that is not known or solvable.  It is usually denoted by a mathematical alphabet ($x$, $y$, $z$, or $\alpha$, $\beta$, etc…).  It is important to distinguish between a variable and an indeterminate in that a variable is solvable, at least conditionally.  To make this more precise, let’s see two examples:

1. 1.

   Let $x$ be a variable such that  $2+3x=a+bx$, where $a,b\in\mathbb{Q}$. Then  $x=(a-2)/(3-b)$.  Here $x$ is solvable conditioned on the equation given. Any values of $a$ and $b\,(\neq 3)$ will yield a value for $x$.

2. 2.

   Let $x$ be an indeterminate such that  $2+3x=a+bx$, where  $a,\,b\in\mathbb{Q}$.  Since $x$ can not be solved, we have  $2=a$  and  $3=b$.  Note that if $a$ and $b$ are previously assigned to be values other than 2 and 3 respectively, then $x$ is no longer an indeterminate.