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 Example

The equation:

$x + 3 y + 2 z = 7$

is a linear equation in $3$ variables.

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.

Also see

  • Results about linear equations can be found here.