inconsistent system

Some demands simply cannot be met together. If one rule insists a number's double is 6 while another insists the same number's double is 10, no number obeys both — the rules contradict. An inconsistent system is exactly this: a system with no solution, because its equations conflict.

When you try to solve an inconsistent system by elimination or substitution, the variables vanish and you are left with a false statement, such as 0 = 7. That impossible equality is the algebraic signal that no values can satisfy the system. There is nothing to find; the solution set is empty.

Geometrically, two inconsistent linear equations are parallel lines with different intercepts — they march in the same direction but never touch. The honest takeaway is that getting no solution is itself a complete and correct answer; an inconsistent system is not a mistake, just a system whose conditions cannot coexist.