Answer
$${\bf{z}} = \left\langle {\frac{7}{2},3,\frac{5}{2}} \right\rangle $$
Work Step by Step
$$\eqalign{
& {\text{Let }}{\bf{u}} = \left\langle {1,2,3} \right\rangle ,{\text{ }}{\bf{v}} = \left\langle {2,2, - 1} \right\rangle ,{\text{ }}{\bf{w}} = \left\langle {4,0, - 4} \right\rangle \cr
& {\text{2}}{\bf{z}} - 3{\bf{u}} = {\bf{w}} \cr
& {\text{2}}{\bf{z}} - 3\left\langle {1,2,3} \right\rangle = \left\langle {4,0, - 4} \right\rangle \cr
& {\text{Solve and simplify}} \cr
& {\text{2}}{\bf{z}} - \left\langle {3,6,9} \right\rangle = \left\langle {4,0, - 4} \right\rangle \cr
& {\text{2}}{\bf{z}} = \left\langle {4,0, - 4} \right\rangle + \left\langle {3,6,9} \right\rangle \cr
& {\text{2}}{\bf{z}} = \left\langle {7,6,5} \right\rangle \cr
& {\bf{z}} = \frac{1}{2}\left\langle {7,6,5} \right\rangle \cr
& {\bf{z}} = \left\langle {\frac{7}{2},3,\frac{5}{2}} \right\rangle \cr} $$
