Solve System of Equations $$2x-y+3z+w=6$$ and $$4x-2y-z+2w=10$$
By eliminating $x,y,w$ we get $z=\frac{2}{7}$
Also we get
$$x=\frac{y}{2}-\frac{w}{2}+\frac{18}{7}$$
$$y=1y+0w+0$$
$$z=0y+0w+\frac{2}{7}$$ $$w=0y+1w+0$$
So we the solution vector as:
$$\begin{pmatrix} x\\ y\\ z\\ w \end{pmatrix}=y\begin{pmatrix} \frac{1}{2}\\ 1\\ 0\\ 0 \end{pmatrix}+w\begin{pmatrix} \frac{-1}{2}\\ 0\\ 0\\ 1 \end{pmatrix}+\begin{pmatrix} \frac{18}{7}\\ 0\\ \frac{2}{7}\\ 0 \end{pmatrix}$$
Now which one we should select as Basis vectors?
The solutions of these equations don't form a vector space, but an affine space (a translated linear space).
A base for the linear space part is the two vectors that $y$ and $w$ are multiplied with. But is that what was asked? You gave a full parametrisation of all solutions (haven't checked the computations, but I'll assume it's OK) and that's enough.