I taught this topic to my students per the textbook, and about half the class was really struggling. Now, most of these students were OK on solving these systems using equations with x, y and z. Why were they not getting the matrix method?
I stopped and rethought the whole enterprise, and realized that the textbook introduced a few rules, without really explaining them, and then jumped straight into somewhat complex problems to solve. Why not follow Carl Menger, and break everything down to the simplest possible elements, and build up from there? So I spent half a class walking the students through the following twelve systems, asking them to write the matrix, and solve it if need be. (Some of the systems are already in the solved form, which was part of the point of the exercise.)
1) x = 7
2) 2x = 14
3) x = 7, y = 4
4) 2x = 14, 2y = 8
5) x + y = 7, y = 4
6) x + y = 7, 2x + y = 10
7) x + y = 7, x + 2y = 11
8) x + y = 7, 2x + 4y = 22
9) 2x + 2y = 14, 2x + 4y = 22
10) x = 3, y = 4, z = 2
11) x + 2z = 7, y + z = 6, z = 2
12) x + 2z = 7, 2y + 2z = 6, y + 3z = 10
Every single student got it. A couple of them broke into big smiles at around equation 8 or 9. One told me, "I had studied this several times before, and I thought I'd never understand it."
So why aren't math textbooks written this way? We teach young kids arithmetic in this fashion: start from 1 + 1 = 2, and work systematically up from there. But at a certain point, it seems to me, math textbooks instead take on a "sort the wheat from the chaff" approach: present the material in a very elliptical fashion, and those who can't get it that way, well, they shouldn't keep going in mathematics!
Me, I think every person on earth with a normally functioning brain can grasp any piece of established mathematics as long as it is presented systematically, with every step along the way explained with simple examples. (And of course, having the creative genius to actually advance mathematics is an entirely different matter.)