"By inspection" means you take a look and try to guess values that will work. In this case I realized I could use as a "wildcard", by setting alpha to 1, I could then set y to whatever I needed to make it work with the values I chose for x and z. Then I tried a few values of x and z that might work, by rewriting
. Since the solution set is positive integers, z > x, so I chose z = 3, x = 2.