umm.. when I subtracted a negative 10.22 with a 19.36, isn't it supposed to be negative 29.58 and not 9.14?

I think this is what you mean

$$\\--10.22+19.36\\
=19.36--10.22\\
=19.36+10.22\\
=29.58$$

 That is positive 29.58

You can think of negative numbers as cold numbers

If you take away cold from a room then the room gets warmer.  

Therefore   $$$--$ can be replaced with a +$$

Yes, subtracting a negative number with a positive number means you keep the negative in this case.