Oh well, what can you do? Sounds like you presented the numbers, she just chose to ignore basic math.
- Apologize to GF for making her feel unloved
- Ask if she'd be willing to talk how she felt about the situation (NOT how you felt)
- Listen to what she says
- Affirm you care about her
- Ask if she'd be willing to hear your side
Issue was resolved one way or another.
But, am I not entitled to feel unloved and disrespected for not even listening to my advice?
All of you keep falling on the same side of the balance
Yes, you are entitled to feel disrespected for how she excluded you from this decision. But you get what you give: your fundamental disrespect for her just bleeds through these posts. Even today, the "trying to put some sense into her" comment -- that starts from the baseline assumption that she has no sense, which is extremely disrespectful. If we can all see that so clearly, what do you want to bet she can, too? And how much do you think that makes her want to open up and invite you into these decisions?
From the way you describe the situation, it sounds like she started off trying to speak your language -- you asked her about her thinking, and she started off with the numbers, because you are math guy, not emotions guy, and so she knew you would be most interested in that. And so you engaged on that basis, and didn't make a similar effort to talk in her language -- instead of acknowledging the reasonableness of wanting a different car and trying to draw out her thinking, you decided it was your role to persuade her that she was wrong. That is not respectful. Respect = listening, asking questions about what *she* thinks, really hearing the things that matter to her, and then asking whether she would like your input and how. Basically, treating her as a competent human being who is capable of making a reasonable decision on her own.
I do think you can do this -- for example, you clearly noticed a similarly emotional response with her and her mother, so you are reasonably attuned to the emotional aspects of her behavior. That's your opening. Instead of "this is a bad choice because math," how do you think the conversation might have gone if you had opened up with something like this: "wow, this really seems important to you. What is it about your car you don't like? [Mm-hmm, mm-hmm] What do you like about the Soul? [Mm-hmm, mm-hmm] What did the guys tell you about this particular one? [Mm-hmm, oh, really? (signal slight skepticism)] Now, I'm just wondering here, so let me know if I'm crossing a line, but I'm a little concerned about the rush to make such a quick decision. Of course, it's totally your call, but I've noticed that it really bothers you that your mother makes impulsive decisions that she regrets later, and I wouldn't want you to end up in the same place just because that's what your mom taught you is normal. What do you want to do? I know you're tight on time -- would you like me to help with some research on what a fair price might be for that Soul?" Etc.
The larger point is that you don't persuade people by telling them the right answer [yes, I know, we are all blatantly violating that rule here]. What you do is lay the arguments piece by piece,
but leave it to them to reach the desired conclusion. That's because if you can convince someone that your answer was
their idea, they will completely commit to it and defend it to the death -- whereas if you try to force it on them, they will dig in their heels. [This is a secret lawyer trick: never ask the ultimate question ("so doesn't that mean") -- instead, you ask all of the smaller questions that make it clear that there is only one possible solution, but you then let the jury make that final connection.]
Also, to address RSM's point, let me reiterate that I think this was a pretty colossally stupid decision on your GF's part. I am jumping on you solely because
you are the one who wrote in asking for advice. If your GF had written in saying my BF totally disrespected me and gave me no end of shit over buying a stupid used Soul on XXX terms, I'd be saying, "look, yes, your BF was a total doink in how he addressed you -- but, damn, that was really a dumb financial move, so maybe you should work on your impulsivity and learn to think things through a little more, because even if you don't care about the money, if you're going to be part of a team, you need to care about his priorities too."