In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables. For example,
{ 3 x + 2 y − z = 1 2 x − 2 y + 4 z = − 2 − x + 1 2 y − z = 0 {\displaystyle {\begin{cases}3x+2y-z=1\\2x-2y+4z=-2\\-x+{\frac {1}{2}}y-z=0\end{cases}}}
is a system of three equations in the three variables x, y, z. A 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 ordered triple
( x , y , z ) = ( 1 , − 2 , − 2 ) , {\displaystyle (x,y,z)=(1,-2,-2),}
since it makes all three equations valid. Linear systems are a fundamental part of linear algebra, a subject used in most modern mathematics. Computational algorithms 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 system of non-linear equations can often be 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 coefficients and solutions of the equations are constrained to be real or complex numbers, but the theory and algorithms apply to coefficients and solutions in any field. For other algebraic structures, other theories have been developed. For coefficients and solutions in an integral domain, such as the ring of integers, 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.
Elementary examples
Trivial example The system of one equation in one unknown
2 x = 4 {\displaystyle 2x=4}
has the solution
x = 2. {\displaystyle x=2.}
However, most interesting linear systems have at least two equations.
Simple nontrivial example The simplest kind of nontrivial linear system involves two equations and two variables:
2 x + 3 y = 6 4 x + 9 y = 15 . {\displaystyle {\begin{alignedat}{5}2x&&\;+\;&&3y&&\;=\;&&6&\\4x&&\;+\;&&9y&&\;=\;&&15&.\end{alignedat}}}
One method for solving such a system is as follows. First, solve the top equation for x {\displaystyle x} in terms of y {\displaystyle y} :
x = 3 − 3 2 y . {\displaystyle x=3-{\frac {3}{2}}y.}
Now substitute this expression for x into the bottom equation:
4 ( 3 − 3 2 y ) + 9 y = 15. {\displaystyle 4\left(3-{\frac {3}{2}}y\right)+9y=15.}
This results in a single equation involving only the variable y {\displaystyle y} . Solving gives y = 1 {\displaystyle y=1} , and substituting this back into the equation for x {\displaystyle x} yields x = 3 2 {\displaystyle x={\frac {3}{2}}} . This method generalizes to systems with additional variables (see "elimination of variables" below, or the article on elementary algebra).
General form A general system of m linear equations with n unknowns and coefficients can be written as
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