The author of the guide is trying to teach some elements of compositional style, emphasizing the idea that mathematics should be presented in a way that is readable as sensible English sentences. From this point of view, the two-column display reads as an alternation of declaratives and imperatives (i.e., statements and commands); substituting "expression" and "unknown" for the precise formulas and writing out the equal sign as the verbal "is equal to," we have the following piece of prose:
A complicated expression is equal to negative one. Solve this
equation. The complicated expression plus one is equal to zero.
Collect the terms on one side. The square of a simpler expression is
equal to zero. Factor. A very simple exponential is equal to one. Use
the Zero Factor Property. The unknown is equal to one. Solve for the
Compare this with the guide's recommended presentation excerpted in Matthew Daly's answer. Among the defects of the two-column presentation, most of the commands (Collect! Factor! Use! Solve!) come after the command has been carried out; only "Solve this equation" precedes the steps that do what that sentence commands. One can, of course, observe that that's not how the imperatives are meant to be understood, but that's sort of the author's point: good, readable, mathematical writing does not contravene the rules of good English (or whatever language is being written in).