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Algebra Systems of Equations (cont’d) Classification of Systems There are two main classifications of systems of equations: Consistent vs. Inconsistent, and Dependent vs. Independent. Consistent vs. Inconsistent •
Consistent Systems have one or more solutions. •
Inconsistent Systems have no solutions. When you try to solve an inconsistent set of equations, you often get to a point where you have an impossible statement, such as “1 2.” This indicates that there is no solution to the system. Dependent vs. Independent •
Linearly Dependent Systems have an infinite number of solutions. In Linear Algebra, a system is linearly dependent if there is a set of real numbers (not all zero) that, when they are multiplied by the equations in the system and the results are added, the final result is zero. •
Linearly Independent Systems have at most one solution. In Linear Algebra, a system is linearly independent if it is not linearly dependent. Note: some textbooks indicate that an independent system must have a solution. This is not correct; they can have no solutions (see the middle example below). For more on this, see the next page. Examples One Solution Consistent Independent No Solution Inconsistent Independent Infinite Solutions Consistent Dependent Version 2.5
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