System of Linear Equation Using Determinants

A system of linear equations (or linear system) is a system in mathematics, that is a collection of one or more linear equations involving the same set of variables. Let us say, in a linear system there are three equations in three variables x, y, z. An assignment of values to the variables such that all the equations are simultaneously satisfied is a solution to the linear system. It makes all three equations valid. Equations are to be considered collectively indicated by the word "system", rather than individually.

The basis and a fundamental part of linear algebra in mathematics, is the theory of linear systems, a subject that is used in most parts of modern mathematics. To find the solutions computational algorithms are an important part of numerical linear algebra, and in engineering, physics, chemistry, computer science, and economics it plays a prominent role. A nonlinear equation system can often be approximated by a linear system (linearization), which is a helpful technique when making a computer simulation or a mathematical model of a relatively complex system.

Often, the coefficients of the equations are complex or real numbers, and the solutions are searched in the same set of numbers. For coefficients and solutions in any field, the theory and the algorithms are applicable. In an integral domain like the ring of the integers, or in other algebraic structures for solutions, other theories have been developed. A method is a collection of Integer linear programming which is used for finding the "best" integer solution (when there are many). A theory based on Gröbner provides algorithms when coefficients and unknowns are polynomials. Tropical geometry is also an example of linear algebra in a more exotic structure.

Solution set

An assignment of values to the variables x1, x2, ..., xn is a solution to a linear system such that each of the equations is satisfied. The solution set is the set of all possible solutions.

The behaviour of a linear system can be any one of the below three possibilities.

1. The system has a single unique solution.

2. The system has infinitely many solutions.

3. The system has no solution.

Geometric Interpretation

In a two-variable (x and y) system, each linear equation determines a line on the xy-plane. Since a linear system solution must satisfy all of the equations, the solution set is the intersection of these lines and is hence either a line, a single point, or the empty set.

Each linear equation for three variables determines a plane in three-dimensional space, and the solution set is the intersection of these planes. Thus a plane, a line, a single point, or the empty set may be the solution set. As an example, the three parallel planes do not have a common point, the solution set of their equations is empty. The solution set of the equations is a single point if three planes intersect at a point, the equations have at least two common solutions if the three planes pass through two points. The solution set is infinite and consists in fact in all the lines passing through these points.

Each linear equation defines a hyperplane in n-dimensional space. The intersection of these hyperplanes is the solution set and is flat, which may have any dimension lower than n.

General Behaviour 

Generally, the behavior of a linear system is determined by the relationship between the number of unknowns and the number of equations. Here, the means of "in general" is a different behaviour that may occur for specific values of the coefficients of the equations.

• A system with fewer equations than unknowns has infinitely many solutions in general, but it may have no solution. This type of system is known as an underdetermined system.

• In general, a system with the same number of unknowns as that of the equations, has a single unique solution.

• Generally, a system with more equations than unknowns has no solution. This type of system is also known as an overdetermined system.

The dimension of the solution set in the first case, is in general, equal to n − m, where n is the number of variables and m is the number of equations.

The trichotomy in the case of two variables is illustrated by the following pictures:

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Many solutions are there of the first system, namely all of the points on the blue line. A single unique solution is there to the second system, namely the intersection of the two lines. There is no solution to the third system since all three lines share no common point. Take note that the pictures above show only the most common case (the general case). It is possible for a system of two unknowns and two equations to have no solution (if the two lines are parallel), or for a system of two unknowns and three equations to be solvable (if the three lines intersect at a single point).