Indeterminate system

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An indeterminate system is a system of simultaneous equations (especially linear equations) which has infinitely many solutions or no solutions at all.

1 Situations
2 Identifying indeterminate systems
3 Useable information
4 See also
5 Sources

[edit] Situations

Any system which has fewer unique equations than variables is indeterminate. 'Unique' equations are equations which cannot be algebraicly derived from each other (especially by scaling, addition, or subtraction.) A common subset of this situation is when two or more linear equations actually describe the same line, plane, or higher dimensional space. For example:

$\begin{cases} 10 = 2x + 4y + 6z \\ 5 = x + 2y + 3z \\ 1 = 5x + 7y + 11z \end{cases}$

is indeterminate. The other common way to look at this problem is that two of the equations co-exist in space and thus intersect at infinitely many points.

Another type of indeterminate system is one that has no solution. In other words, no set of numbers satisfies all of the equations. This is the case with two parallel, but not co-existent, lines or planes. It also can happen when three or more function intersect only in pairs. One such system is:

$\begin{cases} 10 = 2x + 4y + 6z \\ 5 = 2x + 4y + 6z \\ 11 = 8x + 4y + 6z \end{cases}$

where the first two equations can never intersect, and thus no solution exists.

An indeterminate system does not have to be one of linear equations. It could include more complex equations. However, the subject is most commonly explained and significant in linear algebra.

[edit] Identifying indeterminate systems

For linear equations, an indeterminate equation is most easily seen in an augmented matrix. These are some common ways to identify an indeterminate system before gaussian elimination.

If two rows are clearly multiples of one-another, then only one is a unique equation.
If there are fewer unique equations than variables (one less than the number of columns) then the system must be indeterminate.
If there is a nonsense statement in the matrix, where all coefficients are zero but the right-hand value is non-zero, then the system must be indeterminate. This is sufficient at any point in the manipulation of the matrix. Note that this does not apply in reverse; if the coefficients are non-zero and the right hand element is zero, the system may still have a unique solution (or infinitely many unique solutions.)
If two rows of the matrix have identical or scaled coefficients but the right-side entry is not corespondingly scaled or identical, then the matrix is inconsistent and thus indeterminate.

[edit] Useable information

When there are no solutions to a system, its solution set is said to be the empty set.

When a system is underdetermined (has infinitely many solutions,) a common technique is to leave some variables 'free.' Generally, the variable(s) that were not pivot entries after the gaussian elimination of a linear system are used. Then, all of the other variables are defined in terms of the free variable(s). This still offers infinitely many solutions, but it provides some constraints and specificity for those solutions.