Definition:Linear Equation

A linear equation is an equation in the form:

$$y = a_1 x_1 + a_2 x_2 + \cdots + a_n x_n$$

where all of $$a_1, \ldots, a_n, x_1, \ldots x_n, y$$ are elements of a given field.

The point is that all the indices of the $$x$$ and $$y$$ terms in such an equation are $$1$$.

Example
The equation $$y = ax_1^2 + bx_2^2 + cx_3^2$$ is not linear because all the $$x$$ terms are squared.