What is "linear equation"

Linear \Lin"e*ar\ (l[i^]n"[-e]*[~e]r), a. [L. linearis, linearius, fr. linea line: cf. F. lin['e]aire. See 3d Line.]

1. Of or pertaining to a line; consisting of lines; in a straight direction; lineal.

2. (Bot.) Like a line; narrow; of the same breadth throughout, except at the extremities; as, a linear leaf.

3. Thinking in a step-by-step analytical and logical fashion; contrasted with holistic, i.e. thinking in terms of complex interrelated patterns; as, linear thinkers.

   Linear thinkers concluded that by taking the world apart, the actions of people were more predictable and controllable.
   --David Morris (Conference presentation, Fairfield University, October 31, 1997)

   Linear differential equation (Math.), an equation which is of the first degree, when the expression which is equated to zero is regarded as a function of the dependent variable and its differential coefficients.

   Linear equation (Math.), an equation of the first degree between two variables; -- so called because every such equation may be considered as representing a right line.

   Linear measure, the measurement of length.

   Linear numbers (Math.), such numbers as have relation to length only: such is a number which represents one side of a plane figure. If the plane figure is square, the linear figure is called a root.

   Linear problem (Geom.), a problem which may be solved geometrically by the use of right lines alone.

   Linear transformation (Alg.), a change of variables where each variable is replaced by a function of the first degree in the new variable.

n. (context mathematics English) A polynomial equation of the first degree (such as x = 2y - 7)

n. a polynomial equation of the first degree

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. A simple example of a linear equation with only one variable, , may be written in the form: , where and are constants and . 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, , , and , is given by: , where , and are constants and , and 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 , where and are constants and . It has two variables, and , and is non-linear because the sum of the exponents of the variables in the first term, , is two.

This article considers the case of a single equation for which one searches the real solutions. All its content applies for complex solutions and, more generally for linear equations with coefficients and solutions in any field.