What he's talking about when he says "by arithmetic" is what they mean by solving something by inspection. I can tell by just looking at it that x is 4 in the equation 16-x =12. I don't need to "subtract 16 from both sides" and then "divide both sides by negative 1" to solve for x. I can do it by inspection rather than by plodding, pedestrian, algebraic steps. The problem is that as equations get bigger and more complicated, you have to use these mindless algebraic steps. You can't see intuitively what x is by "using arithmetic." You need the mechanics and the discipline of algebra. It's actually really cool to see it work (to solve word problems, for example) sometimes, especially when you'd have no fucking clue what x might be and then algebra works like magic. It's a powerful tool we use, not because we "don't understand what we're doing" (i.e. deducing an unknown's value), but because the task is far too big for our intuition and "inspection" alone.