Unique solution of an overdetermined system

[Shaybay92]

If I want to know how many solutions a consistent linear system with more equations than unknowns has, how do I tell? Obviously there is either 1 solution of infinite solutions. Can you have a free variable in this case? I'm confused how to find out whether a system will give a unique solution.

 

[HallsofIvy]

You need to determine how many independent equations there are. If the system really is consistent, then there must be no more independent equations than unknown variables. That is, if n is the number of variables and m is the number of independent equations, then $n\ge m$. The number of free variables is n- m.

If you write the coefficient matrix for the system and row-reduce, the number of independent equations is the number of non-zero rows.