Definition:Linear Equation

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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 $1$.


Arbitrary Non-Linear Equation

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