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Have a face-to-face conversation. Offer her financial assistance. Be willing to provide physical assistance.
Your first step should be to try to work out a solution to your living arrangements by talking face-to-face. Be honest with your girlfriend about wanting her to move out. If there are problems in your relationship, chances are she feels this tension too and will not be too surprised to hear that you want her to move out. Discuss reasonable options with her and try to come to a consensus.  Take some time before initiating the conversation to brainstorm a few ideas to present. Think about if she has a friend or family member she could stay with, or if you're willing to continue living together until she finds a new living situation. Don't tell her what to do — rather, offer a few possibilities. Say something like, “I know that we thought that moving in together would be a good idea but it just doesn’t seem to be working. Since the apartment was originally mine, I was thinking it would be best if you were the one to move out.” You will likely need to have a few conversations about how you will proceed. If the first conversation about moving is taking place right when you break up, it will likely be difficult for her to make decisions. Allow her some time to process and come to terms with what is happening. If there are children involved, this is definitely the preferred method. You should consider the best living situation for the kids. There is a huge financial burden in moving. She may need a security deposit on a new apartment and utilities. She may also need a moving truck and some furniture. Since this was a mutual relationship, it might be unfair to lay the entire financial obligation of a separation on your girlfriend’s shoulders. Consider splitting the financial costs of moving.  To determine if such a financial split is fair or necessary, think about how much the two of you make. If you make the same or more money than her, it might be unreasonable to ask her to bear the entire burden of a move. If she makes considerably more money than you, there may be no need to help financially. Another option is to allow her to continue living with you while she saves up money for a security deposit and other expenses. The act of moving things out can be difficult to deal with. She might even need help to physically move some of the bigger items out (like a couch). She will need manpower to make the move happen. Offer to help her move large pieces of furniture and boxes to her new space.  If you are breaking up, your physical assistance may not be welcomed. If you are on good terms, however, asking a few friends to help out may make things less awkward and speed up the process.