The system has a unique solution because the values of X1, X2 and X3 are known and unique on each pivot.
Let's assume that we have a matrix of order 3x3 and in each column, there is a pivot variable.
A pivot is a leading coefficient in a row of a matrix. In reduced row-echelon form, a pivot is positioned to where the (1) value is located in a reduced matrix.
If the system or the three equations have a pivot in each column of the matrix, then we conclude that there are unique values of X1, X2 and X3 and they are known and unique.
Also note that there is a unique intersection point from these equations on a cartesian plane, and therefore the variables are unique.
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