Step 1: First we need to install all the required libraries

$ sudo apt-get install curl git-core build-essential zlib1g-dev libssl-dev libreadline5-dev

Step 2: Install RVM

$ bash < <( curl http://rvm.beginrescueend.com/releases/rvm-install-head )

Step 3: mint 10 does not have a .bashrc file so one needs to be added and then RVM specific lines added

$ sudo touch .bashrc
$ sudo gedit .bashrc
(Add line to .bashrc file)
if [[ -s "$HOME/.rvm/scripts/rvm" ]]  ; then source "$HOME/.rvm/scripts/rvm" ; fi

Step 4: close the current terminal and open a new one in order to load the new commands

Step 5: verify RVM has been installed by running you should get back a description of the application written by the author

$ rvm notes

Step 6: now you can install the ruby version of your choice, I am using 1.9.2 so it is compatible with Rails 3

$ rvm install 1.9.2

Step 7: once installed you will need to set the default ruby version to be used by RVM

$ rvm --default ruby-1.9.2

Step 8: verify that the version you selected is now accessible via the standard ruby command

$ ruby -v

Step 9: then use the gem command to install the rails version of your choice

$ gem install rails

Step 10: install MySQL

$ sudo apt-get install libmysql-ruby libmysqlclient-dev
$ gem install mysql2