How do you know a solution space contains the origin?

The solution set contains the origin because if you plug $x=y=z=0$ into your equations, each of them becomes $0=0$ which is true.

If you have nonzero constant terms in one or more of your equations, this will not be the case, and you need to offset the span by a known particular solution, if there are any solutions at all.

This is what's called a "homogeneous system of equations" The origin always satisifes the system, just plug in zeros for all the variables and you'll see all equations hold true.

If you don't have zeros on the right hand side then it's harder, you have to row-reduce the augmented matrix of coefficients in that case.