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The given augmented matrices represent linear systems of equations. Using the variables X sub one, X sub two, X sub three, and so on, identify the basic variables, also referred to as the leading variables and the free variables. We begin by determining the pivots. A pivot in a matrix is the first non-zero entry of each row, when the matrix is in row-echelon form or reduced row-echelon form. So looking at the first augmented matrix, we have two pivots. We have a pivot in row one, column one, as well as row two, column three. And therefore the pivot columns are column one and column three. And now we have the information we need to identify the basic variables, as well as the free variables. A basic variable or leading variable is a variable that corresponds to a column with a pivot, when the matrix is in row-echelon form. Column one is the X sub one column. Column two is the X sub two column. Column three is the X sub three column. And the fourth column is the constant column. And bec
