In the book I am studying I have found the following sentence:
It is often easier to solve the equation by eyeball
What does by eyeball mean?
The word easier is your clue to the eyeball method of the quick scan, using a rough surface assessment instead of any rigor such as algebra.
Example: Say you need two equations, and you have x + y = 3, and 2x + 2y = 6. By eyeball, without math, you might see you have really only one unique equation, and cannot solve the problem yet.
Definition of eyeball:
To measure or estimate roughly by sight: eyeballed the area of the wall that needed paint. American Heritage Dictionary, 5th Ed.
Personal visual inspection, along with the implication of a certain imprecision in the measurement.
In context, I would say it's referring to skipping detailed troubleshooting steps to identify and resolve a problem, and preferring instead to simply have a look and see what's wrong.
This can sometimes work...and sometimes not. For example, the case in 1990 when bolts matched by eyeball were the wrong size, causing an aircraft's windscreen to blow out in flight and the captain to be sucked out of the cockpit (and survive, somehow).
It means that it can be easier to find the solution just by examination, as opposed to following the normal prescribed procedure or solution algorithm.
For example, perhaps you've been taught how to do a u substitution. Sometimes you'll find an integral for which you can just intuit the answer without having to go through the whole rigmarole.
Note, this use of "eyeball" is unusual. It's more common to use "eyeball" (with this same meaning) as a verb, for example, "It might be easier to just eyeball it." However, even this is a bit unusual in the math context.