In this write-up, ["Homogeneous and non-homogeneous equations"](https://www.math.dartmouth.edu/archive/m8s00/public_html/handouts/matrices2/node8.html), it says:

> 
> If the matrix equation
> 
> 
> $\displaystyle AX = B$
> 
> 
> has one particular solution $ X\_p$, and the associated homogeneous
>  equation
> 
> 
> $\displaystyle AX = 0$
> 
> 
> has the complete solution $ X\_h$, then the complete solution to the
>  original non-homogeneous equation is
> 
> 
> $\displaystyle X = X\_p + X\_h.$
> 
> 
> 

**Question:** What I do not understand is the motivation or intuition behind even attempting to look for the nullspace via $\displaystyle AX = 0$ when you already have not just one solution but an infinite number of solutions as the complete solution (via the parameter *t*) of a system of equations where there are more unknowns than equations! *(If needed, please see the Example that immediately follows in the write-up.)*