if you uninstall everything these incantations should work.
#####################################################
# install elixir (only need to do once)
#
# UBUNTU:
#
wget https://packages.erlang-solutions.com/erlang-solutions_1.0_all.deb
sudo dpkg -i erlang-solutions_1.0_all.deb
sudo apt-get update
sudo apt-get install esl-erlang
sudo apt-get install elixir
I’m not confident I can figure out how to uninstall everything, so maybe my best bet is to get asdf working. I followed the instructions on how to do it, and when installing the erlang version it got as far as saying it was missing fop and another tool for building documentation. Usually when I was installing from source this is where the installation ended, but with asdf I waited 5 minutes before killing the process as it seemed like the command would never terminate. I also got stuck on the command
bash ~/.asdf/plugins/nodejs/bin/import-release-team-keyring
it said I was waiting for a key for a long time.
That’s why it’s better to always use a package manager. I also fear that something got broken somewhere during one of your attempts, and even if you end up making it work with asdf or others you might continue experiencing weird things down the line. If I were you I would reinstall the OS (no Arch Linux :p), then either use asdf or the deb from erlang solutions (I think that’s what I did when I was on Mint, and also on debian ; it worked well).
On my machine erlang installation running above 7 minutes - may be you should wait a little longer
I reinstalled Linux Mint version 19 this time, and was able to get Elixir 1.9 working. I tried apt install elixir, but still I the latest version it could find was 1.3.3. I removed it, and removed erlang-base.
What eventually worked was to install erlang by apt install erlang-base and then to download the precompiled elixir 1.9 from the elixir-lang website. (I also tried to compile it myself, but this did not work - there was an error finding something from erlang). I was able to add it my PATH and everything is working smoothly now.
The instructions to install the erlang-solutions .deb file did not work for me - I still only got version 1.3.3 of elixir.
ESL does not build for Linux Mint, so whatever you install seems to assume a version of ubuntu or debian that di not receive any further updates after 1.3.3. As far as I can tell, it assumes Debian Wheezy, as this in fact did not receive the 1.3.4 update from what I see on the ESL download page.
Please do not install random packages that are not meant for your system.
…just joined here in order to clarify a few things for future reference,
hopefully it will save others from similar headaches as per above.
Erlang Solutions’ instructions per se indeed don’t / won’t work on Mint ‘as is’. But…
that has absolutely nothing to do though whatsoever with Mint somehow having a different package base or dependencies etc.
The ‘fault’ lies in their installer’s routine, erlang-solutions_1.0_all.deb…that’s what’s to be blamed 
What it does, is that it parsers /etc/issue for the identification of the numerous Ubuntu & Debian versions,
then nicely sets the repositories accordingly to the version detected.
But…by parsing the non-standard /etc/issue, which will 99% return a different id,
in any Ubuntu / Debian derivative out there…it fails on such.
A by far more proper way would be if they checked /etc/lsb-release.
Actually, even better than such, if we are to go with the freedesktop standards nowadays, /etc/os-release instead.
Although as we all understand…admittedly even then some distros would fail to comply, lol…
Hopefully Erlang Solutions will revisit their os detection routine at some point,
at least without relying entirely on /etc/issue…
For the time being, on Ubuntu derivatives, someone can ‘bypass’ the erlang-solutions_1.0_all.deb method,
by following these instructions here…
https://forums.linuxmint.com/viewtopic.php?p=1603722#p1603722
That’s for older Mint 18 (xenial). For Mint 19 (bionic), obviously the line should instead be ubuntu bionic contrib…
…bend wasn’t the first one to be hit by this, that’s what i’m trying to say
Just noticed his thread over at LM’s forums today…
where people also told him to use the erlang-solutions.deb,
apparently unaware of what goes on. Had i seen his thread earlier,
i’d have posted / replied there, but maybe that’s for the better…
Thought i’d mention the above here, so in case some other fella in the future,
from whatever other Ubuntu / Debian derivative comes asking,
can probably understand what’s going on & save his/her time…
Or so that you guys around here can guide him on what to do to easily add the repos & packages properly:
all that’s needed is to add the appropriate matching repo in sources.list
Or maybe the ES people eventually notice it, and then fine-tune / somehow tweak the routine in their ‘setup’ .deb… 
