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 $1$.
The equation $b = a_1 x_1^2 + a_2 x_2^2 + a_3 x_3^2$ is not linear because all the $x$ terms are squared.