I hope this is going toward Cauldwell learning a lesson—I use reciprocals to solve math and science problems all the time. (Including unit analysis, the go-to quick check that you have set up a problem correctly.) Your math-based education is grinding to a very early halt if you stick at “rewrite an algebra equation? Sorry, my calculator can’t do that.”
Now inverse matrices: that’s something that, if you use them in real life, you use a computer. I’d have sympathy if he wanted to skip those. (Though if you use them in real life, you would also be expected to go through the exercise by hand early on, so it’s not all “the magic box says the number is 3.46, and that’s all I understand.” If you want to apply things in new ways, you need a deeper understanding than “the magic box says.”)
Caulfield, I mean.