Understanding Linear Systems: Unique, Inconsistent, and Infinitely Many Solutions

Three Possibilities for a Linear System

1. There is no solution2. There is exactly one solution3. There are infinitely many solutions

A linear system is a set of linear equations that must be solved simultaneously to find the unknowns. There are three possibilities for the nature of the solution of a linear system:

1. Unique Solution: If the system of linear equations has one and only one solution, it is called a unique solution. This means that there is only one set of values for the unknowns that satisfies all the equations in the system. Geometrically, this can be represented by the intersection of the lines or planes that represent the equations in the system.

2. No Solution: If the system of linear equations has no solution, it is called an inconsistent system. This means that there is no set of values for the unknowns that satisfies all the equations in the system. Geometrically, this can be represented by the parallel lines or planes that represent the equations in the system.

3. Infinitely Many Solutions: If the system of linear equations has more than one solution, and these solutions form a straight line or a plane, it is called an infinitely many solutions system. This means that there are infinitely many sets of values for the unknowns that satisfy all the equations in the system. Geometrically, this can be represented by the coincident lines or planes that represent the equations in the system.

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