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Free vs Dependent Variables and Vectors

Systems of linear equations sometimes have one solution (unique), no solution (an inconsistent system), or infinitely many solutions. In the last case, there are free variables in the system that can take on any numerical value while the other variables can be successfully solved given those settings. This video connects the ideas of finding these solutions to a geometric representation as lines or planes (or their generalizations) based on a given point on the solution set (a particular solution) and direction vectors that come from the free variables.