Systems of linear equations

Each linear equation in three unknowns describes a plane. Solving a system means finding the points common to all of them, and that geometric picture explains the whole answer. Three planes can meet at a single point, or in a common line, or share a whole plane, or have no common point at all — and there is no arrangement giving exactly two intersection points. That is why the count of solutions is always zero, one, or infinity, never anything else.

Counting equations does not settle which case you are in. Three equations in three unknowns look decisive, but if one equation is just the sum of the other two it adds no information and the solution set is a line. Conversely, a system with more equations than unknowns is usually inconsistent but need not be — the extra equations may simply repeat what is already known. The reliable test is the determinant of the coefficient matrix: non-zero means exactly one solution, zero means the transformation collapses space and the answer is either none or infinitely many.

Gaussian elimination is the method that settles it in practice. Add multiples of rows to other rows to introduce zeros below the diagonal, then read the answer back by substitution. Its virtue is that it does not need to know in advance which case applies: a row of zeros equalling zero reveals a redundant equation and infinitely many solutions, while a row of zeros equalling a non-zero number is the contradiction 0 = 1 and proves there is no solution.