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Well if you are asking when a system of linear equations has a single solution, it would be when the rank of the matrix of coefficients is equal to the rank of the augmented matrix.
However this can also lead to infinite solutions, so you do not want a row with all zeroes. (ie two or more rows are not linearly independant.)
liner systems are having only 1st order variables, thus leads to single solutions. if there will be "N" order in a system then it will lead for N number of solutions.
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