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I am working on this system of equations, and I do not get the same results as they appear in the final solution in the book, need your assistance here...this is the question:

Discuss the solutions of the equation system:

\[\begin{matrix} ax_{1}+bx_{2}+2x_{3}=1\\ ax_{1}+(2b-1)x_{2}+3x_{3}=1\\ ax_{1}+bx_{2}+(b+3)x_{3}=2b-1 \end{matrix}\]

I have applied elementary row operations R2->R2-R1 and R3->R3-R1 and got this matrix:

\[\begin{pmatrix} a &b &2 &1 \\ 0 &b-1 &1 &0 \\ 0 &0 &b+1 &2b-2 \end{pmatrix}\]

However, in the book they say the matrix after elementary row operations is:

\[\begin{pmatrix} a &1 &1 &1 \\ 0 &b-1 &1 &0 \\ 0 &0 &b+1 &2(b-1) \end{pmatrix}\]

which is odd since I see no reason to touch the first row.

This is not the end of the troubles, the solution to the problem according to the book is:

"There are six cases:

b=1: infinite solution

b=5, a=0: infinite solution

b=5, a~=0: unique solution

b=-1: no solution

b~=+1 or -1 or 5, a~=0: unique solution

b~=1 or 5, a=0: no solution

(~= means not equal, gave up figuring it out in latex)

Where did they get the 5 from and where and how shall I see this in the matrix ?

Thanks !