Is it better to drive or fly to Alaska from Chicago?

I have a potential job lined up in Anchorage, and I currently live in Chicago. I have a 1996 Jeep Cherokee that I would like to drive up, compared to flying. I do have a dog (pit bull) that is moving with me and many airlines don't let them fly (she could easily pass as a mix). I'm a vet tech and understand what animals go through in flight and I do want to bring my bike and more luggage then what I could carry on to a plane. I know gas to drive there is expensive but I feel like the costs of flight, transporting a pet, and having to rent a car would even out. I have my passport and know I need all my dog's records to cross border. What does everyone think?