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If I have a system of N linear equations with N unknowns, how do I know if each unknown has atleast one solution. I havn't taken any formal classes in linear algebra, so please if you use any abstract terminology associated with linear algebra, explain it please. I would appreciate any help.

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This is a useful method: en.wikipedia.org/wiki/Gaussian_elimination I don't think there is any simple way of going around linear algebra on a question like this (assuming that your N equations are of linear form). –  Thomas E. Nov 27 '12 at 9:23
    
I apologise I should have reworded my question better. I don't need help solving a system of linear equations, I want to know under what condinitions a system with n unknowns and n equations IS solveable. I am also not asking you to not use linear algebra, I am just asking you to explain any abstract notation, if you chose to use it. –  Ethan Nov 27 '12 at 9:26
    
Are we to assume these are linear equations? I think in general knowing in advance whether we can solve any set of $N$ equations in $N$ unknowns is very difficult, if not impossible. However if they are linear equations then there is a very simple test, calculate the determinant (en.wikipedia.org/wiki/Determinant). If it is non-zero (which is extremely likely) then you have a unique solution. If it is $0$ then you either have no solutions or infinitely many, and it is easy to check which is the case. –  fretty Nov 27 '12 at 9:32
    
Yes all the equations are linear, and I don't even need unique solutions, I just need to know that all N unknowns have atleast one solution. Also I don't know what a "determinant" is, if you could help me out, id appreciate it. –  Ethan Nov 27 '12 at 9:38
    
no you can't always find solution, the coefficients must satisfy certain conditions –  user31280 Nov 27 '12 at 10:09
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  • Take a look at this link explaining where a certain matrix is proven to be non invertivible using a system of linear equations.
  • Also read this Wikipedia article explaining the properties of a system of linear equations.
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