System of linear equations

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a system of linear equations (or linear system) is a collection of two or more <a href="/facts/Linear_equation/JfCgALwf">linear equations</a> involving the same <a href="/facts/Variable_(math)/VbQlrvKz">variables</a>.
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 <a href="/facts/Solution_(mathematics)/2RKaXrDM">solution</a> 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 <a href="/facts/Tuple/BI1HIxL8">ordered triple</a>

(
        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 <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, a subject used in most modern mathematics. Computational <a href="/facts/Algorithm/fnl5NmRt">algorithms</a> for finding the solutions are an important part of <a href="/facts/Numerical_linear_algebra/jnS6Tteb">numerical linear algebra</a>, and play a prominent role in <a href="/facts/Engineering/RFVg8H3I">engineering</a>, <a href="/facts/Physics/a7aH2y08">physics</a>, <a href="/facts/Chemistry/SrNqkGVm">chemistry</a>, <a href="/facts/Computer_science/lN3pEyPz">computer science</a>, and <a href="/facts/Economics/HYctA2Dn">economics</a>. A <a href="/facts/Nonlinear_system/4aYItALC">system of non-linear equations</a> can often be <a href="/facts/Approximation/LvYHsXzq">approximated</a> by a linear system (see <a href="/facts/Linearization/ohYSeQR3">linearization</a>), a helpful technique when making a <a href="/facts/Mathematical_model/CGhyuxgz">mathematical model</a> or <a href="/facts/Computer_simulation/Re6o8vzN">computer simulation</a> of a relatively <a href="/facts/Complex_system/eGG3B1nn">complex system</a>.
Very often, and in this article, the <a href="/facts/Coefficient/5Hwq2Wuq">coefficients</a> and solutions of the equations are constrained to be <a href="/facts/Real_number/R02gw5Pb">real</a> or <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>, but the theory and algorithms apply to coefficients and solutions in any <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>. For other <a href="/facts/Algebraic_structure/TkuXca24">algebraic structures</a>, other theories have been developed. For coefficients and solutions in an <a href="/facts/Integral_domain/cylt6zxW">integral domain</a>, such as the <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> of <a href="/facts/Integer/PUwwrawx">integers</a>, see <a href="/facts/Linear_equation_over_a_ring/2rGTE3GK">Linear equation over a ring</a>. For coefficients and solutions that are polynomials, see <a href="/facts/Gr%C3%B6bner_basis/efMrjM7w">Gröbner basis</a>. For finding the "best" integer solutions among many, see <a href="/facts/Integer_linear_programming/MDaaadoA">Integer linear programming</a>. For an example of a more exotic structure to which linear algebra can be applied, see <a href="/facts/Tropical_geometry/fFy8Evgv">Tropical geometry</a>.

System of linear equations open-in-new

System of linear equations