What is m if system has unique solution (`alpha,beta,theta` )? 2x+y-mz=1 x-y+z=-1 3x+2y+z=2

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The problem provides the information that the system has unique solution, hence, by Cramer's rule, the determinant of square matrix of coefficients of unknowns needs not to be zero, such that:

`Delta = [(2,1,-m),(1,-1,1),(3,2,1)] != 0`

Evaluating the determinant, yields:

`Delta = [(2,1,-m),(1,-1,1),(3,2,1)] ` `= -2 - 2m + 3 - 3m - 4 - 1`

`Delta = -5m - 4 != 0 => -5m != 4 => m != -4/5`

Hence, evaluating the valid values of `m` , under the given conditions, yields that `m in R - {-4/5}.`

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