Solving linear equations involving many variables

Question

I know that to solve a linear equation involving $n$ variables for example, we need $n-1$ other independent equations to form a system and then solve that system. Is there a formal proof for this?

Answer 1

Proof: if we didn't have n equations for n unknowns, then the matrix representing the system would either be rectangular (meaning no unique solution) or have zero rows (also no unique solutions). Therefore there must be n equations for n unknowns in order for a unique solution to exist.

Answer 2

If it's a system of linear equations, then you can use Gauss-Jordan elimination to get the solutions. You can also take the determinant of the coefficient matrix first to check whether or not you'll have a single solution, infinitely many solutions, or zero solutions.