How are these two equal?(equation, inequality)

[Pithikos]

How are these two equal??(equation, inequality)

I study discrete mathematics and we are doing combinations at the moment. There is this example in the book(Discrete Mathematics and Combinatorics p. 30 Ex. 1.43) where it states that the number of integer solutions for:

x₁+x₂+x₃+x₄+x₅+x₆<10 where x_i\geq0

is equal to the number of integer solutions of

x₁+x₂+x₃+x₄+x₅+x₆+x₇=10 where x_i\geq0 and x₇>0

Can someone explain me this? The author supposes that I magically understand what goes through his mind.

Here is a screenshot of the problem: http://img251.imageshack.us/img251/823/garbageab.jpg

 

[saim_]

This is subtle, but, trivial: Try to see that to all the possbile solutions of x₁+x₂+x₃+x₄+x₅+x₆<10, we can add a positive number and make it equal to 10 and also that it is only the set of these solutions to which we can add a positive number and make it equal to 10; thus the set of integer solutions of both equations have the same number of elements.