What is defined as a system of equations or inequalities that has no solution?

A system of equations or inequalities that has no solution is referred to as an inconsistent system. This type of system occurs when the equations represent lines (or planes in higher dimensions) that do not intersect at any point. In the case of two linear equations, for example, if they are parallel, they will never meet, leading to no solutions.

An inconsistent system can also apply to inequalities when no values satisfy all the inequalities simultaneously. In general, identifying an inconsistent system involves recognizing that the equations or inequalities are contradictory, leading to a situation where no value can satisfy all conditions set by the system.

The other definitions do not apply to situations with no solutions. An independent system does have a solution that is unique, while a dependent system has infinite solutions. A homogeneous system, on the other hand, typically consists of equations set to zero, which can also have non-trivial solutions depending on the context, but doesn't inherently imply that there are no solutions available.