Solving a set of 2 equations and two unknowns by determinants is trivial. Solving 3, 4 and 5 equations with 3, 4 and 5 unknowns is a real pain to do by hand and I had to do that.
Well, it was pretty routine if you went to college in the 60's or before (i.e., before calculators and personal computers existed). (Note I'm not being condescending! It's still manual, boring labor.) I'm still a bit taken aback by how calculators have destroyed the arithmetic and elementary plotting abilities of folks of later generations.
Here's an example. My daughter took me to lunch yesterday and wanted to leave a nice tip for the waitress. She intuitively left $5 on a bill that was slightly over $30. She asked me if that was a good tip, then proceeded to use her cell phone to figure out the percentage. Long before she had even typed it in, I told her it was a 16.7% tip. She looked at me amazed and wondered how I did that. 5/30 is 1/6 and I know most of the decimal expansions of fractions from having used them so many times. There was no trick -- just experience.
Another example: another child unit and I were talking and a multiplication like 43*17 came up. I spoke up with the answer, 731, and was looked upon as some savant. How ridiculous! Every child is taught the distributive law, so this is easily rewritten as 43*(10 + 7) = 430 + 301. Even if you can't keep the least significant digits in your head, you can do the major parts and get around 730. When we had to do calculations without calculators, you of course learned these "tricks" or you didn't survive in a technical career.
Another area most kids (and I'm speaking of kids that are, say, under 40 or so) fail in is the ability to generate plots quickly. If there's no computer or plotting calculator around, they're dead in the water. It's a shame they don't learn and practice plotting things by hand. You learn to plot the key points in functions (look for zeros and poles) and, again, approximate the behaviors as necessary. Then sketch the curve; this used to be part of every basic calculus class (do they do this anymore?). Another area is when analyzing a bunch of data from an experiment -- it's pretty easy to grab some graph paper and make a quicky plot. There are all kinds of plots that can be made: a scatter plot, stem and leaf, probability, box and whisker, histogram, etc. While it's nice to use a computer if you can, you shouldn't hesitate to use a hand-plotted graph if no automation is present. Experienced engineers/scientists quickly learn that plotting your data is one of the most important things you can do.
Geez, this turned into a sermon from the old geezer...