In the context of systems of equations, the terms "inconsistent," "dependent," and "neither" refer to the relationships between the equations and their solution sets.
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Inconsistent: A system of equations is called inconsistent if there are no solutions. This occurs when the equations represent lines (in two dimensions) or planes (in three dimensions) that do not intersect at any point. For example, two parallel lines in a 2D plane are inconsistent because they will never meet.
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Dependent: A system of equations is considered dependent if there are infinitely many solutions. This occurs when the equations represent the same line, plane, or hyperplane in their respective dimensions. In graphical terms, two lines that lie on top of each other are dependent; they intersect at every point along the line.
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Neither (or Independent): A system is labeled neither inconsistent nor dependent when it is independent, meaning there is exactly one unique solution. In this case, the equations represent lines that intersect at exactly one point (in two dimensions) or planes that intersect at a single line or point (in three dimensions). This type implies that the equations are distinct and not parallel or overlapping.
In summary:
- Inconsistent: No solutions, equations do not intersect.
- Dependent: Infinitely many solutions, equations coincide.
- Neither: Exactly one solution, equations intersect at a single point.
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