Unique, inconsistent and dependent systems

**After elimination** | **Interpretation** | **Action** Every variable is fixed | One solution | Substitute to verify it. An impossible row, such as 0=5 | No solution; the system is inconsistent. | Identify the contradictory equation. At least one free variable and no contradiction | Infinitely many solutions | Choose parameters and express all variables in terms of them. Example For x+y+z=4, 2x+2y+2z=8 and x-y+z=0, the second equation repeats the first. Subtracting the third from the first gives 2y=4, so y=2 and x+z=2. The full solution is (x,y,z)=(2-t,2,t), t in R. A single choice such as (2,2,0) does not describe all solutions. Replacing the second equation by 2x+2y+2z=9 gives 0=1 after subtracting twice the first equation, so there is no solution. Technology check A row of zeros in a reduced matrix indicates a redundant equation; a zero coefficient row with a nonzero final entry indicates inconsistency. A solver error alone does not distinguish these cases. Inspect the reduced equations or perform elimination, and report the mathematical conclusion.