solutions theorem

[bugzpodder]

is there a theorem that says that if you have n linear equations, each equation is distinct and contains all of the n variables, can you say that there always exist at least one solution?

[Guv]

There is a theorem with some caveats that says there is always a solution for a set of n equations in n unknowns. Each variable need not explicitly appear in every equation. For example, consider the following system.

x + y = 3
x + z = 4
y + z = 5

It is equivalent to the following system.

x + y + 0*z = 3
x + 0*y + z = 4
0*x + y + z = 5

The caveats in the theorem relate to ambiguous or inconsistent systems.

An ambiguous system can have many valid solutions. If any equation is a linear combination of one or more of the other equations, you have ambiguity. A very trivial system with this property is the following.

x + y = 2
2x + 2y = 4

Anybody would recognize that there is somthing wrong with the above system. This problem can be far less obvious. For example, make up nine equations in ten unknowns. Then multiply equation one by 3, subtract equation two, add twice equation 3, add equation 4, subtract equation 5. Use the result as the tenth equation. This system is ambiguous. You are not likely to notice a problem until you try to solve such a system.

A system can be inconsistent, in which case there is no solution possible. A very trivial and obvious example is the following.

x + y = 7
x + y = 8

There are more complex systems which are inconsistent, but which look okay until you try to solve them.