A hammer and a nail cost $1.10, and the hammer costs one dollar more than the nail. How much does the nail cost? The answer is 0.05. I used algebra. But my friends say, why not 0.10, and they also say, it can be that the hammer is 1.04 and the nail 0.06. How do I tell them that 0.05 is the definite  answer, nothing else; I need mathematical evidence and proof to convince folks. 

So, (1) Hammer+Nail =$1.10 
(2) Hammer-Nail = $1

From equation (1)
Let Hammer = 1.10 - nail
So,
Substitute Hammer into equation 2:
1.10 - nail - nail = 1
1.10-1 = 2 nails
0.1    = 2 nails
1 nail = $0.05
