So today my mom found out I wanted to apply to Wellesley and Swarthmore college and a couple other LACs that are less competitive and she wasn’t happy to say the least… She never went to college but my sister was the first person in my family to go to college and she went to Drexel U in Pennsylvania. Now she expects me to go to a large research university like she did but the thing is I really want to go to a liberal arts college. I’ve tried to explain to her several times why I want to go to a LAC but she doesn’t seem to care and believes their only schools for people who want a degree in something like film, dance, art, etc. She also complains that we won’t be able to afford to pay for these schools, which I completely understand since we’re definitely not rich, but most of the schools I’m applying to meet full need and I know that they either give either better or at least equal financial aid to what my sister receives at Drexel (I’ve used the FA calculators on the websites). I’ve also applied to a couple state schools in NY and I qualify for free tuition so if we really can’t afford a private college I have a backup plan. She believes that I won’t be able to get into any of the schools and says I’m wasting my time. I realize Wellesley and Swarthmore are reaches but the rest of the colleges on my list are target and safety schools and I think I have a pretty good chance of getting into some of them. I really don’t know what to do because she’s helping me pay for college and this is becoming a reoccurring argument. If anyone can give me some advice please share.
Tell her you won’t go to those schools unless they are in budget and show all the schools you are willing to attend that will be in budget. Show the majors at the schools as well.
Hmm. Can you show her the websites with the list of majors? Things like math, physics, chem, bio, and econ are also liberal arts majors. Many have CS programs, too.
Ask her to run the net price calculators on each college website with you, too. So she can see that they probably offer better aid than Drexel.
Additionally, LACs don’t have grad students. So your classes are mostly taught by professors. If you are in a field that requires undergrad research, often you get better access at LACs because there are no grad students to help the profs, so undergrads get all the opportunities.
And LACs are very good prep for grad school: This is a very good piece by a scientist who graduated from Grinnell:
http://www.thecollegesolution.com/wp-content/uploads/2012/01/cech_article2.pdf
There are also a lot of fsmous grads from the schools you mentioned. Hillary Clinton, Madeleine Albright, Cokie Roberts, and Diane Sawyer are all Wellesley alums. Mike Dukakis, Jonathan Franzen, and James Michener are all Swat grads (and 2 of my cousins graduated from there - one is an MD, and the other a physics professor).
If you are thinking of going to grad school, you might show her this. Swat figures prominently on the lists:
http://www.thecollegesolution.com/the-colleges-where-phds-get-their-start/
Finally, can you get your GC or a more knowledgeable family member to talk to get?
Here’s something else you may want to show your mom.
https://www.cbsnews.com/news/where-professors-send-their-children-to-college/
You could also try working out a compromise with your mom such as she will tour a LAC with you and in exchange you will tour a college of her choice. She may come around when forced to take a closer look.
You can also give her a “parent pick” school where you will apply only if she backs off when you round out the rest of your list.
Swat and Wellesley are great schools. I agree with you that your mom should give them a fair hearing, with the caveat you will only go if the FA makes it affordable.
Your inital post implies your Mom is happy with how things turned out for your older sister at Drexel. If that is the case can you get your sister involved in the conversation?
Note that this scientist went on to win a Nobel Prize in chemistry. So he’s got cred. 
Talk to your mom and try to find out what the real issue is:
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Is it future employment? Drexel is known for engineering and has a coop program. She may be worried with a liberal arts degree that you won’t be employable.
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Is it location? Drexel is in Philly but Swarthmore is in PA, so that doesn’t seem to be an issue.
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Is it cost? Ask for a budget and say that you know you need to make your college meet that budget.
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Is it realistic for you to get into Swarthmore? Maybe she is trying to save you heartbreak from not getting in.
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What is it about a LAC that is appealing to you?
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Are the app fees an issue? can you get a waiver?
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Have you/her run the Net Price Calculators for some schools to see what financial aid you would be likely to get?
I would suggest saying something like “Mom, I will be happy to put a Large Research University on my list. Help me pick one out that has X, Y and Z qualities. I also understand that college budget is an issue. Give me a number and I will choose a college that meets that budget. But right now, I would like to apply to a variety of colleges and see where I get in. I will only choose two reaches as I understand that applications fees are expensive. I understand your concerns about employability and I will make sure to do internships in (major) to prepare for that.”
Drexel is outrageously expensive and not known for offering financial aid so not sure I follow the reasoning about it being cheaper. Also, I would not exactly call Drexel a large research institution in the sense of a Penn State or an Ivy. It does what it does well which is prepare people professionally for careers - it’s all pre-professional at the grad level, too. Maybe that’s what your mom appreciates about it.
I went to an LAC and, yes, back when US News and World Report was about where profs sent their kids, LACs had a great moment, but they truly function primarily to prepare students for grad school. So I can see where your mom might have concerns - she is not entirely unjustified. Maybe you can compromise and look at some LAC-like Honors Programs at some large research universities which will be amazing experiences. My well-respected LAC just started an “honors” program where they have students read articles and discuss them which is absurd and disappointing and not a mark of a stellar, challenging education. LACs are scrambling right now to pull in students and then deliver on a promise get students career-ready so they are a little in flux and destabilized. I see large research institutions as being a more stable bet for academic opportunities and for networking/access to helpful professors. A small school might only have a few professors in your field.
Of course, if you get into Wellesley or Swarthmore, you will want to go for those!