Editing System of linear equations (section)

{{Short description|Several equations of degree 1 to be solved simultaneously}}
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[[File:Secretsharing 3-point.svg|thumb|A linear system in three variables determines a collection of [[plane (mathematics)|planes]]. The intersection point is the solution.]]

In [[mathematics]], a '''system of linear equations''' (or '''linear system''') is a collection of two or more [[linear equation]]s involving the same [[variable (math)|variable]]s.{{sfnmp |1a1= Anton |1y=1987 |1p=2 |2a1=Burden |2a2=Faires |2y=1993 |2p=324 |3a1=Golub |3a2=Van Loan |3y=1996 |3p=87 |4a1=Harper |4y=1976 |4p=57 }}<ref>{{cite encyclopedia |url=https://www.britannica.com/science/system-of-equations |title=System of Equations |encyclopedia=Britannica |access-date=August 26, 2024 }}</ref>
For example,
: <math>\begin{cases}
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\frac{1}{2}y-z=0
\end{cases}</math>
is a system of three equations in the three variables {{math|''x'', ''y'', ''z''}}. A ''[[Solution (mathematics)|solution]]'' to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. In the example above, a solution is given by the [[Tuple|ordered triple]]
<math>(x,y,z)=(1,-2,-2),</math>
since it makes all three equations valid.

Linear systems are a fundamental part of [[linear algebra]], a subject used in most modern mathematics. Computational [[algorithm]]s for finding the solutions are an important part of [[numerical linear algebra]], and play a prominent role in [[engineering]], [[physics]], [[chemistry]], [[computer science]], and [[economics]]. A [[Nonlinear system|system of non-linear equations]] can often be [[Approximation|approximated]] by a linear system (see [[linearization]]), a helpful technique when making a [[mathematical model]] or [[computer simulation]] of a relatively [[complex system]].

Very often, and in this article, the [[coefficient]]s and solutions of the equations are constrained to be [[Real number|real]] or [[complex number]]s, but the theory and algorithms apply to coefficients and solutions in any [[field (mathematics)|field]]. For other [[algebraic structure]]s, other theories have been developed. For coefficients and solutions in an [[integral domain]], such as the [[Ring (mathematics)|ring]] of [[integer]]s, see [[Linear equation over a ring]]. For coefficients and solutions that are polynomials, see [[Gröbner basis]]. For finding the "best" integer solutions among many, see [[Integer linear programming]]. For an example of a more exotic structure to which linear algebra can be applied, see [[Tropical geometry]].