System of linear equations

In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables. For example,

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

since it makes all three equations valid.

In mathematics, the theory of linear systems is a branch of linear algebra, a subject which is fundamental to modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and such methods 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.

Properties

Independence

The equations of a linear system are independent if none of the equations can be derived algebraically from the others. When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set. For linear equations, logical independence is the same as linear independence.

For example, the equations

are not independent — they are the same equation when scaled by a factor of two, and they would produce identical graphs. This is an example of equivalence in a system of linear equations.

For a more complicated example, the equations

are not independent, because the third equation is the sum of the other two. Indeed, any one of these equations can be derived from the other two, and any one of the equations can be removed without affecting the solution set. The graphs of these equations are three lines that intersect at a single point.

The equations x − 2y = −1, 3x + 5y = 8, and 4x + 3y = 7 are not linearly independent.

Consistency

The equations 3x + 2y = 6 and 3x + 2y = 12 are inconsistent.
The equations of a linear system are consistent if they possess a common solution, and inconsistent otherwise. When the equations are inconsistent, it is possible to derive a contradiction from the equations, such as the statement that 1 = 3.

For example, the equations

are inconsistent. In attempting to find a solution, we tacitly assume that there is a solution; that is, we assume that the value of x in the first equation must be the same as the value of x in the second equation (the same is assumed to simultaneously be true for the value of y in both equations). Applying the substitution property (for 3x+2y) yields the equation 6 = 12, which is a false statement. This therefore contradicts our assumption that the system had a solution and we conclude that our assumption was false; that is, the system in fact has no solution. The graphs of these equations on the xy-plane are a pair of parallel lines.

It is possible for three linear equations to be inconsistent, even though any two of the equations are consistent together. For example, the equations

are inconsistent. Adding the first two equations together gives 3x + 2y = 2, which can be subtracted from the third equation to yield 0 = 1. Note that any two of these equations have a common solution. The same phenomenon can occur for any number of equations.

In general, inconsistencies occur if the left-hand sides of the equations in a system are linearly dependent, and the constant terms do not satisfy the dependence relation. A system of equations whose left-hand sides are linearly independent is always consistent.

The equations 3x + 2y = 6 and 3x + 2y = 12 are inconsistent.

Equivalence

Two linear systems using the same set of variables are equivalent if each of the equations in the second system can be derived algebraically from the equations in the first system, and vice-versa. Equivalent systems convey precisely the same information about the values of the variables. In particular, two linear systems are equivalent if and only if they have the same solution set.

Reference:
http://en.wikipedia.org/wiki/System_of_linear_equations