# Linear equation

In elementary algebra, linear equations are algebraic equations in which both sides of the equation are polynomials or monomials of the first degree - i.e. each term does not have any variables to a power other than one.

## Form and Connection to Analytic Geometry

In general, a linear equation with $n$ variables can be written in the form $\sum_{i=1}^{n}a_ib_i=c$, where $a_i$ is a series of constants, $b_i$ is a series of variables, and $c$ is a constant.

In other words, a linear equation is an equation that can be written in the form $$a_1b_1 + a_2b_2 + ... +a_nb_n = c$$, where $a_1, a_2,... , a_n$ are constants multiplied by variables $b_1, b_2, ..., b_n$ and $c$ is a constant.

For the particular case $n=1$ (single variable equation), the resulting equation can be graphed as a point on the number line, and for the case $n=2$ (resulting in a linear function), it can be graphed as a line on the Cartesian plane, hence the term "linear" equation. This can extended to a general Cartesian n-space, in which the linear equation with the corresponding number of variables can be graphed as an n-1-space - this concept is the idea behind analytic geometry as envisioned by Fermat and Descartes.