Relation Between Solvability and Multiplicity
Given any system of linear equations, there is a relation between multiplicity and solvability.
If one equation is a multiple of the other (or, more generally, a sum of multiples of the other equations), then the system of linear equations is undetermined, meaning that the system has infinitely many solutions. Example:
has solutions for such as (1, 1), (0, 2), (1.8, 0.2), (4, −2), (−3000.75, 3002.75), and so on.
When the multiplicity is only partial (meaning that for example, only the left hand sides of the equations are multiples, while the right hand sides are not or not by the same number) then the system is unsolvable. For example, in
the second equation yields that which is in contradiction with the first equation. Such a system is also called inconsistent in the language of linear algebra. When trying to solve a system of linear equations it is generally a good idea to check if one equation is a multiple of the other. If this is precisely so, the solution cannot be uniquely determined. If this is only partially so, the solution does not exist.
This, however, does not mean that the equations must be multiples of each other to have a solution, as shown in the sections above; in other words: multiplicity in a system of linear equations is not a necessary condition for solvability.
Read more about this topic: Elementary Algebra, Solving Algebraic Equations
Famous quotes containing the words relation between, relation and/or multiplicity:
“There is a certain standard of grace and beauty which consists in a certain relation between our nature, such as it is, weak or strong, and the thing which pleases us. Whatever is formed according to this standard pleases us, be it house, song, discourse, verse, prose, woman, birds, rivers, trees, room, dress, and so on. Whatever is not made according to this standard displeases those who have good taste.”
—Blaise Pascal (1623–1662)
“There is a constant in the average American imagination and taste, for which the past must be preserved and celebrated in full-scale authentic copy; a philosophy of immortality as duplication. It dominates the relation with the self, with the past, not infrequently with the present, always with History and, even, with the European tradition.”
—Umberto Eco (b. 1932)
“Is a Bill of Rights a security for [religious liberty]? If there were but one sect in America, a Bill of Rights would be a small protection for liberty.... Freedom derives from a multiplicity of sects, which pervade America, and which is the best and only security for religious liberty in any society. For where there is such a variety of sects, there cannot be a majority of any one sect to oppress and persecute the rest.”
—James Madison (1751–1836)