# Consisity of non-homogenous systems of linear equations

Consistency of non-homogenous systems of linear equations

Hello,

Is it possible to find out if a non-homogenous system of linear equations has:
- Single Solution.
or
- Infinite Solutions.
or
- No Solutions.
Without applying any methods/rules ( Such as Gauss, determinant..etc), just judging by the values and number of given equations and unknowns?

Thanks for your time.

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## Answers and Replies

HallsofIvy
Homework Helper
I'm not sure what you mean by "judging by the values" etc. All of the methods you want to "not apply" do just that. If you mean "just by looking at the values (of the coefficients?) and number of given equations and unknowns" without doing any computation at all, then, except for the obvious: if there are fewer equations than unknowns, there cannot be a unique solution (but can be no solution or an infinite number of solutions) but if you had the same number of equations and unknowns or more equations than unknowns, no solution, a unique solution, or an infinite number of solutions are all possible depending on the determinant of coefficients.

Well, ya I meant the values of the coefficients. To be more clear, the question was I encountered was:
Without solving, determine which of the following linear systems have a solution or not"
...list of systems goes here..

I guess I'm allowed to calculate the determinant after all as it doesn't solve the system by itself..

Well, I read this on an online preview of some book:
Nonhomogeneous system:
(a) If det(A) $$\neq$$ 0, a unique solution exists.
(b) If det(A) = 0, either no solutions exists or infinitely many solution exist.

So even the determinant of the matrix doesn't classify the system into either: Has solution or Has no solution. Is their is such a rule/method that can do that without solving the system?

Thanks for your response and I apologize for the ambiguity.

HallsofIvy