Intersection of 3 planes in 3D space

The key thing you pointed out is that $Ax = b$ having a unique solution for some $b$ shows that the columns of $A$ are linearly independent. In particular, the columns of $A$ are three linearly independent vectors in $\mathbb{R}^3$, so they actually form a full basis for $\mathbb{R}^3$. By the definition of a basis, every vector in $\mathbb{R}^3$, such as the vector $(2,3,4)$, can be uniquely written as a linear combination of $A$'s columns. This unique linear combination corresponds to the unique solution to $Ax = (2,3,4)$, so the answer is (B).

The values of $\alpha_i$ are irrelevant, only that the planes intersect at a unique point. The coefficient matrix is thus of full rank, and for any given $\alpha_i$ said matrix can be inverted and left-multiplied with the $\alpha_i$ vector to yield a unique solution for $x,y,z$. Thus B is the correct answer.