Unique, infinite, and no solutions

Three possibilities

• A system of three linear equations has exactly one of these outcomes:
  • One (unique) solution — a single triple $(x, y, z)$ works.
  • Infinitely many solutions — a whole family of triples works.
  • No solution — no triple works at all.
• Each linear equation in three unknowns is a plane. The outcome depends on how the three planes sit:
  • They meet at a single point → one solution.
  • They share a whole line (or coincide) → infinitely many.
  • They have no common point (e.g. two are parallel, or they form a triangular "prism") → no solution.

What elimination reveals

• Carry out elimination as normal. The final line tells you the type of system:
  • A definite value such as $z = 4$ → unique solution.
  • Everything cancels to a true statement, $0 = 0$ → infinitely many solutions. The equations are dependent (one carries no new information).
  • Everything cancels to a false statement, $0 = k$ with $k \ne 0$ → no solution. The equations are inconsistent.

Consistent and inconsistent

• Consistent — at least one solution exists (unique or infinite).
• Inconsistent — no solution exists ($0 = k$ appears).
• Dependent — an equation is a combination of the others, so it adds nothing new (leads to infinitely many).

Describing an infinite solution set

• Choose the "free" variable and call it a parameter, e.g. $z = t$.
• Express the other variables in terms of $t$.
• Write the general solution as a triple, e.g. $(x, y, z) = (t - 2,\; 8 - 2t,\; t)$.
• Every value of $t$ gives one valid solution — hence infinitely many.