Arguing for Christians in relationships?

From some studying the bible, it says not to argue, quarrel, all that. Well, my girlfriend and I are both Christians, and like all relationships we argue. Recently I have been trying not to argue back with her and hold my tongue because most time I argue due to trying to prove myself right, pride, and all of that. However, I feel that since I don't argue back, or try not to, our problems can't be solved. How is a Christian to go about these problems? Try to stay out of the arguement and then talk about the problem peacefully? Please, serious answers only. And scoffers, I love ya guys, but I'm trying to get some serious answers.