System of Linear Equations

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A System of Linear Equations is a equation system of linear equations.

  • AKA: SLE.
  • Context:
  • Example(s):
    • [math]\displaystyle{ \begin{array}{lcl} 2x_1-x_2 & = & 1 \\ x_1+x_2 & = &2 \end{array} }[/math]. This is a consistent system with unique solution [math]\displaystyle{ x_1 = 1, x_2 = 1. }[/math]

    • [math]\displaystyle{ \begin{array}{lcl} x_1+x_2 & = & 2 \\ 2x_1+2x_2 & = &4 \end{array} }[/math]. This is a consistent system with infinitely many solutions. One of the solution of the system is [math]\displaystyle{ x_1 = 1, x_2 = 1. }[/math]

    • [math]\displaystyle{ \begin{array}{lcl} x_1+x_2 & = & 2 \\ x_1+x_2 & = &1 \end{array} }[/math]. This is an inconsistent system. So no solution exist for the system.

    • [math]\displaystyle{ \begin{array}{lcl} x_1+x_2+x_3 & = & 3 \\ x_1-x_2-x_3 & = & 1 \\ x_1+2x_2+3x_3 & = & 4 \end{array} }[/math]. This system is consistent with a unique solution [math]\displaystyle{ x_1 = 2, x_2 = 1, x_3= 0 . }[/math]

    • [math]\displaystyle{ \begin{alignat}{7} 3x &&\; + \;&& 2y &&\; - \;&& z &&\; = \;&& 1 & \\ 2x &&\; - \;&& 2y &&\; + \;&& 4z &&\; = \;&& -2 & \\ -x &&\; + \;&& \tfrac{1}{2} y &&\; - \;&& z &&\; = \;&& 0 & \end{alignat}. }[/math]This system is consistent with a unique solution [math]\displaystyle{ x_1 = 1, x_2 = -2, x_3= -2 . }[/math]
  • Counter-Example(s):
  • See: Linear Function, Formal System, Mathematical Model, Linear Algebra, Equation Solving, Numerical Linear Algebra.


References

2015

  1. The subject of this article is basic in mathematics, and is treated in a lot of textbooks. Among them, Lay 2005, Meyer 2001, and Strang 2005 contain the material of this article.

2011

2007