Upgrading from 2009 to 2010 is not possible. The bugs and updates page says the following:
It is not possible to upgrade from TL 2009 to TL 2010, so install the new release in a new directory (as is done by default). install-tl may offer to import options (not packages) from a previous release, but results are not guaranteed.
On Ubuntu 11.04, a Debian derivative, and Tex Live installed in /home/mas/texlive on an x86_64 system then the following additions at the end of ~/.profile:
PATH=/home/mas/texlive/2011/bin/x86_64-linux:/home/mas/bin/:$PATH; export PATH
MANPATH=/home/mas/texlive/2011/texmf/doc/man:$MANPATH; export MANPATH
INFOPATH=/home/mas/texlive/2011/texmf/doc/info:$INFOPATH; export INFOPATH
get TeX Live working without a problem, including latexmk from the command line.
I have the Ubuntu TeX packages installed as well but, where the same command exists in /home/mas/texlive/2011/bin/x86_64-linux (or in /home/mas/bin/ for some manually-installed software) then that version will be used in preference.
I am using a locally-installed download of TeXworks with the modification you show: In Edit -> Preferences -> Paths frame the path '/home/mas/texlive/2011/bin/x86_64-linux' has been put at the top of the list.
Coming on to latex-mk. This is not in the default list for TeXworks processing tools, although latexmk is (as LaTeXmk on my systems as they are not the latest versions: see the question latexmk-not-accompanying-texworks-0-5 for the change that TeXworks not longer lists latexmk in recent builds). If you want latex-mk and have it installed (it is not part of TeX Live but will probably be in the repositories for many GNU/Linux distributions) then it can be added to the list of processing tools. In TeXworks I added it via Edit -> Preferences in the Processing Tools frame, click '+' to add and fill in the new window appropriately (I used 'LaTeX-mk' for name, 'latex-mk' for program (no path needed unless you have more than one installed and don't want the default), and options '--pdflatex' followed by '$fullname') and that seems to behave reasonably. If you wanted to add any other processing tools the procedure is similar, though the parameters will probably be different, of course.
Best Answer
Open your .bashrc script and write the following lines at the end:
PATH=/usr/local/texlive/2018/bin/x86_64-linux:$PATH; export PATH MANPATH=/usr/local/texlive/2018/texmf-dist/doc/man:$MANPATH; export MANPATH INFOPATH=/usr/local/texlive/2018/texmf-dist/doc/info:$INFOPATH; export INFOPATHAnd then you should be good to go.