# System of Linear Equations

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

**AKA:**SLE.**Context:**- It can range from being a Consistent SLE (if it has a solution) to being an Inconsistent SLE.
- It can have unique solution or infinitely many solutions if it is consistent.
- It can be the subject area of a Linear Algebra Subject Area.
- It can be an input to a Linear Equation System Solving Task (and have a system of linear equations solution such as a general solution of the SLE)

**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]

- [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]
**Counter-Example(s):****See:**Linear Function, Formal System, Mathematical Model, Linear Algebra, Equation Solving, Numerical Linear Algebra.

## References

### 2015

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/System_of_linear_equations Retrieved:2015-11-8.
- In mathematics, a
**system of linear equations**(or linear system) is a collection of linear equations involving the same set of variables.^{[1]}For example, : [math]\displaystyle{ \begin{alignat}{7} 3x &&\; + \;&& 2y &&\; - \;&& z &&\; = \;&& 1 & \\ 2x &&\; - \;&& 2y &&\; + \;&& 4z &&\; = \;&& -2 & \\ -x &&\; + \;&& \tfrac{1}{2} y &&\; - \;&& z &&\; = \;&& 0 & \end{alignat} }[/math] is a system of three equations in the three variables*x*,*y*,*z*. A**solution**to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by : [math]\displaystyle{ \begin{alignat}{2} x &\,=\,& 1 \\ y &\,=\,& -2 \\ z &\,=\,& -2 \end{alignat} }[/math] since it makes all three equations valid. The word "*system*" indicates that the equations are to be considered collectively, rather than individually.In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of 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, the coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers, but the theory and the algorithms apply for coefficients and solutions in any field. For solutions in an integral domain like the ring of the integers, or in other algebraic structures, other theories have been developed, see Linear equation over a ring. Integer linear programming is a collection of methods for finding the "best" integer solution (when there are many). Gröbner basis theory provides algorithms when coefficients and unknowns are polynomials. Also tropical geometry is an example of linear algebra in a more exotic structure.

- In mathematics, a

- ↑ 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

- Mark V. Sapir http://www.math.vanderbilt.edu/~msapir/msapir/jan10.shtml#system
- QUOTE: A system of linear equations is any sequence of linear equations. A solution of a system of linear equations is any common solution of these equations. A system is called consistent if it has a solution. A general solution of a system of linear equations is a formula which gives all solutions for different values of parameters.
Examples. 1. Consider the system: [math]\displaystyle{ x + y = 7 \\ 2x + 4y = 18 }[/math] This system has just one solution: x=5, y=2. This is a general solution of the system.

- QUOTE: A system of linear equations is any sequence of linear equations. A solution of a system of linear equations is any common solution of these equations. A system is called consistent if it has a solution. A general solution of a system of linear equations is a formula which gives all solutions for different values of parameters.

### 2007

- http://www.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html
- QUOTE: Mathematical Challenge Eight: Beyond Convex Optimization
Can linear algebra be replaced by algebraic geometry in a systematic way?

- QUOTE: Mathematical Challenge Eight: Beyond Convex Optimization