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**SyNtHeSiS** Consider the system of linear equations:

$\displaystyle x + 2y + 3z = 4$

$\displaystyle x + y - 2z = a$

$\displaystyle 2x + 3y + bz = 6$

where a and b are constants

For which values of a and b does this system have (i) a unique solution; (ii) no solution; (iii) infinitely many solutions?

Would you first have to gauss reduce this matrix to get 0s below the first non-zero entry of the first row, second row and third row, and then look at what values of a and b that give the particular solutions? Or would you look for the a and b values that give the particular solutions without gauss reduction?