Can anyone come up with a nice way to balance chemical equations manually? All the web seems to offer is either vague "fiddle around until it works" or the nuclear option "translate into a simultaneous linear system and solve". Is there anything in between? It need not be theoretically optimal, just easy to apply by hand to small solvable cases. I'm imagining a well-defined procedure repeatedly "adjusting" coefficients until "done", then dividing out their common factor, perhaps akin to an n-D raster line drawing algorithm that somehow manages to hill climb onto a scaled solution. It should be more clever than, say, mindlessly trying all the possible cases in some fixed order, yet stay grounded in the problem domain. It might even be "morally equivalent" to Gaussian elimination but performed directly on the chemical equations. Longhand division is kind of like this. There's a little eyeballing and maybe even some backtracking estimating the digits, but it's a reasonably effective way to arrive at the answer by hand. Crunching determinants for simple chemistry is analogous to using Newton's method on everyday division problems. Any ideas?