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In [[mathematics]], or more specifically [[algebra]], a '''linear equation''' is an [[equation]] involving only [[polynomials]] of the first degree.  Linear equations are ubiquitous in [[applied mathematics|application]]s of mathematics.  Equations involving a single variable appear in the simplest problems where one unknown quantity needs to be determined from other given information.  

Quadratic equations occurring in applications typically involve [[real number]] [[coefficient]]s.  However, one can algebraically manipulate polynomial equations in the usual way as long as the coefficients can be added  and multiplied together.  Please see the [[linear equation|main page]] for a discussion of linear equations with real coefficients.

The most general mathematical context that deals with systems of objects that can be added and multiplied together is [[ring (mathematics)|ring]] theory.  One can define polynomials, and in particular quadratic polynomial equations, as long as the coefficients are in a ring.  The real numbers is an example of a ring.  Another example, important in [[coding theory]], is polynomials with coefficients in the ring <math>\mathbb{Z}_2 = \{ \, \overline{0}, \overline{1} \, \}</math>.  You add and multiply in this ring in the same way you add or multiply the integers <math>\{ \, 0, 1, \, \}</math> with one exception:  since <math>\mathbb{Z}_2</math> does not have a "two" in it, we set <math>\overline{1} + \overline{1} = \overline{0}</math>.

== Solutions of linear equations ==

When working with polynomials over a specific ring <math>R</math>, one usually looks for solutions in the same ring <math>R</math>.  The main exception to this is the most common case, where a polynomial has [[integer]] coefficients but one desires real number solutions.  If, instead, one demands solutions of the same type as the polynomial coefficients, namely integers, the equation becomes a [[Diophantine equation]].  In this article, we assume that the desired solutions are in the same ring that the coefficients are drawn from.  

=== Linear equations over a division ring ===

Every non-degenerate linear equation with real coefficients has a solution.  This is no longer the case when the coefficients and solutions are in a general ring, but it is true if the ring is a [[division ring]].  The method of finding solutions looks formally the same as that when the coefficients are real numbers:  after using algebraic operations to isolate one variable, every solution can be found by making an arbitrary choice of values for the remaining variables.  In particular, a linear equation over a division ring with infinitely many elements will have a either zero, one, or infinitely many solutions.

=== Linear matrix equations ===


=== Linear Diophantine equations ===


=== Linear congruences ===