How to get my mom to reconsider getting rid of my dog?

I have three dogs. My sister got a puppy a few months ago(she's over 18) and my mom said she WOULD NOT pay for anything the puppy needs because its not her responsibility(which I understand) My sister unexpectedly moved to another state with her boyfriend and left the puppy here.(I know, that's very dumb of her) .. I've been paying for everything the puppy needs and my sister sends me money for stuff also. The only thing my mom needs to do is take me to go get dog food(which we already have to do for my two older dogs anyways) My mom wants to get rid of Izzy(the puppy) and she is not giving me any reason at all, which I think is unfair. I know that she is the adult and everything, but please help. I'm 16 right now and after high school i'm moving in with my dad(for college) and he has agreed to let me take the dog.. but my mom doesn't want to wait. How can I get her to reconsider or atleast talk to me about it?