Let's say we're asked to solve the following system of equations for the unknowns $x$ and $y$.
$$x+y=1 \qquad \text{(1)}$$
$$x-y=3 \qquad \text{(2)}$$
My approach, which I'll refer to as serial elimination of variables (SEV for convenience) would be to solve one of the equations for one of the unknowns, substitute that in for that variable in the other equation, solve this equation for the other variable, and then back-substitute to get the other unknown. E.g., $x=3+y$, $(3+y)+y=1$, $y=-1$, $x=2$.
Many of my students seem to have been taught a different procedure, which I'll call SEE for setting equations equal. They would solve both equations for a particular variable, set them equal to each other, solve for that variable, and then back-substitute. So faced with system (1,2), they would do $x=1-y$, $x=3+y$, $1-y=3+y$, $y=-1$, $x=2$.
SEE seems correct but slower. I can think of examples where it doesn't work, e.g.,
$$e^x+xy=1 \qquad \text{(3)}$$
$$\sin y+xy=3 \qquad \text{(4)}$$
You can't even get going with (3,4) by the SEE method, whereas with SEV you can at least reduce it to a single equation in a single variable, and maybe look for a solution by graphing.
I also see the following over and over. Say we have the system (5,6):
$$x+y=0 \qquad \text{(5)}$$
$$x-y=7 \qquad \text{(6)}$$
My students proceed by writing $x-y-7=0$, then setting $x+y=x-y-7$ (setting 0 equal to 0). This hasn't helped, and they don't know what to do next. This seems to suggest that SEE has a disadvantage, which is that it obscures what one is trying to do (eliminate a variable), and this obscurity causes problems for students in applying it.
Why is SEE seemingly so popular? Does it have some advantage that I'm not aware of?