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Classification of Systems

There are three possibilities for the manner in which the graphs of two linear equations could meet--the lines could intersect once, not intersect at all (be parallel), or intersect an infinite number of times (in which case the two lines are actually the same).

If the two equations describe the same line, and thus lines that intersect an infinite number of times, the system is dependent and consistent.

If the two equations describe lines that intersect once, the system is independent and consistent.

If the two equations describe parallel lines, and thus lines that do not intersect, the system is independent and inconsistent.

Classification of Systems

Thus, a system is consistent if it has one or more solutions. A system of two equations is dependent if all solutions to one equation are also solutions to the other equation.

The following chart will help determine if an equation is consistent and if an equation is dependent:

Chart of Dependency and Consistency

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