Solve the system
\begin{array}{r@{}r@{}r@{}r@{}r@{}r@{}r@{}r} x_1 & - 2 x_2 & - 2 x_3 & & + 5 x_5 & - 4 x_6 & = & -1 \cr & & & - x_4 & + 4 x_5 & + 2 x_6 & = & 7 \cr x_1 & - 2 x_2 & & & + 9 x_5 & - 6 x_6 & = & 3 \end{array}
\begin{bmatrix} x1 \\ x2 \\ x3 \\ x4 \\ x5 \\ x6 \end{bmatrix} = __ + t1 __ + t2 __ + t3 __ + t4 __ + t5 __
where t1, ..., t5 are all free variables.
Apologies, I'm still trying to figure out how to use LaTeX and don't know how to make t1,...,t5 italicized, or have the [x1,...,x5] inline.
Anyways, I've reduced the matrix to
\begin{Bmatrix}1 & -2 & 0 & 0 & 9 & -6 & 3 \\ 0 & 0 & 1 & 0 & 2 & -5 & 2 \\ 0 & 0 & 0 & 1 & -4 & -2 & -7 \\ \end{Bmatrix}
and I know 3 things that DO belong in the blanks. $$ [x1,...,x6] = \langle 3,0,2,-7,0,0\rangle + t1\langle ,1,0,0,0,0\rangle + t2\langle -9,0,-2,4,1,0\rangle $$ I'm pretty sure one of them should also be $\langle 6,0,5,2,0,1\rangle$, and then the rest are $\langle 0,0,0,0,0,0\rangle$, however that's incorrect and I'm stuck as to why.
Any help is appreciated, thank you