Linear equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable (however, different variables may occur in different terms). A simple example of a linear equation with only one variable, $x$ , may be written in the form: $ax + b = 0$ , where $a$ and $b$ are constants and $a \neq 0$ . The constants may be numbers, parameters, or even non-linear functions of parameters, and the distinction between variables and parameters may depend on the problem (for an example, see linear regression).

Linear equations can have one or more variables. An example of a linear equation with three variables, $x$ , $y$ , and $z$ , is given by: $ax + by + cz + d = 0$ , where $a, b, c$ , and $d$ are constants and $a, b$ , and $c$ are non-zero. Linear equations occur frequently in most subareas of mathematics and especially in applied mathematics. While they arise quite naturally when modeling many phenomena, they are particularly useful since many non-linear equations may be reduced to linear equations by assuming that quantities of interest vary to only a small extent from some "background" state. An equation is linear if the sum of the exponents of the variables of each term is one.

Equations with exponents greater than one are non-linear. An example of a non-linear equation of two variables is $axy + b = 0$ , where $a$ and $b$ are constants and $a \neq 0$ . It has two variables, $x$ and $y$ , and is non-linear because the sum of the exponents of the variables in the first term, $axy$ , is two.

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