Can you solve 2-sat problem when truth assignments of some variables are determined

Question

I am trying to find a assignment to satisfy a 2-sat statement. Problem is some of the clauses are 0 or x or 1 or x. I think the 1 or x clauses have no effect on the solution, but the 0 or x clauses determine the value for x. As I have clauses like ~x or y the determined value for x results in determining the value for y and so on. Can I solve this problem with Krom's procedure?