Let $(O,e_1,e_2,e_3)$ and $(O',e_1',e_2',e_3')$ be two coordinate systems. Let $\overline{OO'}=2e_1-e_2+3e_3$, $e'_1=e_1-e_2+3e_3$, $e'_2=e_1+e_2+e_3$ and $e'_3=e_1-e_2-e_3$.
a) Find the coordinates of a point A in $(O',e_1',e_2',e_3')$ if the coordinates of A in $(O,e_1,e_2,e_3)$ are $(2,3,4)$.
b) Find the coordinates of a point A in $(O,e_1,e_2,e_3)$ if the coordinates of A in $(O',e_1',e_2',e_3')$ are $(2,3,4)$.
Where should I use the information about $\overline{OO'}$? Isn't the answer of a) $(11,9, -5)$?
You have equations expressing $e_i$'s in terms of $e_i$, you need inverse equations expressing $e_i$ in terms of $e_i$'s, so express your equations as a matrix $$\left( \begin{array}{c} e_1'\\ e_2'\\ e_3' \end{array}\right)=\left( \begin{array}{ccc} 1 &-1&3\\ 1&1&1\\ 1&-1&-1 \end{array} \right)\left( \begin{array}{c} e_1\\ e_2\\ e_3 \end{array} \right)$$ Taking the inverse matrix you get $$\left( \begin{array}{c} e_1\\ e_2\\ e_3 \end{array}\right)=\frac{1}{4}\left( \begin{array}{ccc} 0 &2&2\\ -1&2&-1\\ 1&0&-1 \end{array} \right)\left( \begin{array}{c} e_1'\\ e_2'\\ e_3' \end{array} \right)$$ which gives you expressions of $e_i$'s in terms of $e_i'$'s.
Now, part a): you have that the coordinates of A in $(O,e_1,e_2,e_3)$ are $(2,3,4)$, hence $$A=O+2e_1+3e_2+4e_3$$
$$=O'+\overline{O'O}+2e_1+3e_2+4e_3$$ $$=O'-\overline{OO'}++2e_1+3e_2+4e_3$$ $$=O'-(2e_1-e_2+e_3)+2e_1+3e_2+4e_3$$ $$=O'+4e_2+3e_3$$ Now replace $e_2$ and $e_3$ with their expressions in terms of $e_2'$ and $e_3'$ giving $$A=O'-\frac{3}{4}e_1'+2e_2'-\frac{5}{4}e_3'$$ so $A$ has coordinates $(-\frac{3}{4},2,-\frac{5}{4})$ in the coordinate system $(O',e_1',e_2',e_3')$.
Similarly for part b).