Hey - can you please post the part of the explanation you don't understand (and give me some detail on what's troubling you)?

Also, please post the entire question (including answer choices) whenever you post questions. Thanks!

Ok - so you understand how we wrote the first 4 equations:

1) History students: a + b + c + 3 = 25
2) Math students: e + b + d + 3 = 25
3) English students: f + c + d + 3 = 34
4) TOTAL students: a + e + f + b + c + d + 3 = 68

(For others - it helps to look at the diagram in the attached file to write the above 4 equations and to know what each variable represents.)

And you understand that adding the first 3 of the above equations gives you this new equation:
5) a + e + f + 2b +2c +2d + 9 = 84

In the next step, we are not subtracting just the total students from equation #5, but rather we are subtracting entire equation #4 (see above).
So, (entire equation #5) - (entire equation #4) which is the same thing as (history + math + English students) - (total students).

This subtraction gives us the overlap. The overlap is represented by people taking 2 classes and people taking 3 classes, but we have a real number (3) for the people taking 3 classes, so the only overlap variables left in the problem are those variables that represent people taking 2 classes.
eqn #5: a + e + f + 2b +2c +2d + 9 = 84
minus eqn #4: -[a + e + f + b + c + d + 3 = 68]
equals 0 + 0 + 0 + b + c + d + 6 = 16
equals b + c + d = 10

And since b, c, and d represent my three "attends 2 classes" categories, the answer is 10.

Does that make sense?