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 [itex]n\ge m[/itex]. 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.