Well, I agree with your row reduction. However, I'm not sure I follow your back substitution. I would probably substitute in for the higher-indexed variables first. For example, back substitution yields
Generally, to find the basis required, you can't plug in specific values for variables. In the case of this underdetermined system, you're going to have to use parameters. That is, suppose and Can you write the whole solution in terms of these two parameters? If so, what do you suppose the solution space looks like?
Can you continue from here?