When I was growing up and one of us said that something wasn't fair my mother nearly always said, "sorry, but life is unfair." I would get frustrated and determined to give explanations whenever possible.
Some things are managed by rules. Turning off the lights is a rule for the family. A guest doesn't live in the house and therefore doesn't have to follow the family rules. We do treat guests differently than family members. It could be said that they've got the better deal. However, we're all guests at sometime and so it evens out.
In my and my daughter's house, take home leftovers belong to the person who ordered them. We frequently write our name on the container to remind someone that they belong to so and so. But if that's not the rule in your house then you go by whatever rule you have.
Making rules or guidelines for living together can be worked out over time. I prefer having family meetings so that the whole family can have input, especially when a child is school age. Twelve is a difficult age in that the brain development causes the child to pretty much look at things as black and white. It really helps to have them participate in this sort of rule making so that they will feel that things are more fair.
I'm sure she said you played favorites. Geez, that is so common among siblings. They will always believe that mom and dad have a favorite and it's not them. A fact of life. You can make an effort to reduce that feeling but the feeling will always be there.
Your girls are 10 years apart in age and there is no way that you can do the same thing or spend the same amount of money for each one. Don't even try and don't get into that discussion with the oldest one. There really are times that the most appropriate answer is, "I'm sorry you feel that way. Life is just not fair sometimes."
The more you try to convince her that you're being fair the more she'll fight with you about it. Play your, "I'm the parent." card and tell her that you do the best you can to treat both of your girls fairly but life is such that you can't always do it. Talk with her, when she's not complaining, about how you have to give them things differently because of their age differences. Remind her that you paid $200 for her camp and that she got toys instead when she was 2. What you do for each of them is based on their age and needs. That is the way life is.
I suggest that you read the book or listen on CD How To Talk So KIds Will Listen & Listen so Kids Will Talk by Adele Faber and Elaine Mazlish. They've also written a book called Siblings Without Rivalry. How we feel about what someone says depends a lot on how we word what we say.
Accepting the other person's feelings makes a big difference too. It took me years to remember to say something like, "I know you're angry and think this was unfaiir. Now that I think about it, perhaps it was. But this is the best that I could do at the moment." Deal only with the one situation at that time. Do not allow the situation to turn into a "kitchen sink" argument during which she says, "you're always unfair," etc. Say, "we'll talk about this at another time" and then don't respond to whatever she says. If you feel that you have to say something repeat the same words.
And then later, when a particular situation has not caused a discussion, you bring up the subject and talk together about how she feels. Accept what ever she says. Don't argue with her. Tell her you'll listen to her thoughts and ask her to listen to yours. And if you can't reach an agreement then drop the subject with a I'm sorry that you're upset. IF you've gained a better understanding of what's going on with her, tell her that you heard what she said and you're sympathetic but you can't change what is. "Life is unfair."
Don't even try to make things even. My daughter tries to keep track of how many times, hours, nights that both of her children spend with me and from time to time tells me it's not fair because my granddaughter spends more time with me then my grandson. She puts the pressure on me to even it up. I listen and sympathize but I don't try to even it up. I also don't tell her that life is unfair. She knows that but just can't apply it in this situation. I agree with her that one has been here more than the other and that's the end of the conversation. Nothing changes. It's like she draws a line thru that total and starts over with keeping track. Somehow the two of us have silently agreed that this is the way it is and it's OK as long as she reminds me from time to time and I agree.
The bottom line for most of us is that we want to be heard. Try agreeing with your daughter. With practice both of you will let it drop once she's felt heard. She's a teen. You'll need to find someone else to hear you. This sight can help with that.