Linear algebra. Find a base that generates the vector room the equation system describes.

Just an example. I know one can find vectors which is common for the both equations by just looking and them and figuring out, but is there some method that will always work?

If you have j equations in a k-dimensional room, how can you get a base for the sub room you will get (the sub room written in parameter form)? (The sub room will have the dimension k-j, iff no of the equations is a linear combination of the other equations.)