Can we borrow against the equity to pay for independent living?

Isn't - omg yes on looking into a scholarship funds.

Now that I think back on this (1990's), there was $$ in future charges paid for in the initial buy-in that my aunt never used as she died soon after moving in. That $$ I'm remembering was not refundable but went into a general fund @ the CCRC. I bet the state requires that some of this go into a fund for others @ the CCRC. Yes, selling her unit was totally at the mercy of the CCRC as they do the listing and vett the applicants, even though my aunt owned it and had a deed for it. It was a bit of paperwork in probate to work out & really needed having an attorney who understood the situation.

It's unlikely your mom will be able to take out a reverse mortgage or equity loan against the CCRC unit. Buying into a CCRC is not not like traditional real estate purchase. You don't actually 'own' your unit and can't sell it, only the CCRC can. But, once a resident starts to run out of money, many CCRCs allow a them to draw on the 'equity' of their entrance deposit to pay for their monthly fee.
'igloo572' gives good advice - sit down with the director of the CCRC and lay out your mom's finances. If they don't allow the use of the $82K to start subsidizing her monthly fee, there are still a couple possibilities that may work for your mom. They may be able to give her a break on her monthly fee or, many CCRCs have sort of 'scholarship funds' that they can draw on to make up the difference for residents who start to run short.
It also wouldn't hurt to invest in an hour with an elder law attorney before you delve into all this.

No , you can not, but SHE can.

Pug - You really need to carefully review the contract your mom signed. Based on the brief info, I'm thinking she is in a CCRC w/a 82K buy-in? A Continuing Care Retirement Community. For that amount I would imagine she doesn't actually own anything, per se, as 82K is kinda low for a CCRC. Usually if there is no ownership then there isn't anything that is her equity to borrow against.

I had an aunt in a CCRC and her buy in was well mid6 figures but she had ownership via a deed for her unit but there also were fees related to health care costs that would not ever be refundable if she left the CCRC or died before using them fully. CCRC seem to work well if they are pretty well off and still have tax liability as some of the CCRC health care costs are deductions passed down to the individual residents. When she died (I was her executrix), the unit was eventually sold BUT the CCRC had all kinds of covenants regarding to whom the unit could be sold to as they had to qualify financially and health wise to live there. It was an eye-opener for me. Believe me the contract will not be to your mom's best benefit.

If this is a CCRC and she is having problems paying CCRC fees plus her insurance premiums and her personal expenses now while she is still healthy, you might have to have a hard talk with her and the facility to determine if she is really a good long-term resident @ the place especially due to increased costs if her health care needs change and she has to pay extra for all that. Good luck.