\begin{definition}[Definition:Linear Equation]
A '''linear equation''' is an equation in the form:
:$b = a_1 x_1 + a_2 x_2 + \cdots + a_n x_n$
where all of $a_1, \ldots, a_n, x_1, \ldots x_n, b$ are elements of a given field.
The point is that all the indices of the $x$ and $y$ terms in such an equation are $1$.
\end{definition}