rank and reduced row echelon form issue

Question

I have following problem; I have given a matrix (1,2,3; 4,5,6; 7,8,9). it has to be equal (1,1,1) I have done reduced row echelon form and ended up with following; (1,2,3 | 1; 0,3,6 |3; 0,0,0 | 0) so if x3=x3, then we may express x2=3-2x3 x1=x3-5 so if I set x3=0, I end up with x1=-5 and x2=3; however, if I replace these values to the original equation, I see erroneous result. Can someone explain to me what is my mistake/\

Answer 1

With $z=0$ you get $x=-1$ and $y=1$

Your system is dependent because the middle row is the average of the other two rows so you have infinitely many solutions.

For example you may assume $x=0$ and solve for $y$ and $z$ to get $$ (x,y,z)= (0,-1,1)$$

You may let $x=t$ and solve for $y$ and $z$ in terms of $t$

My solution is $$(x,y,z)=(t,-2t-1,t+1)$$