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.
Show your friends that their proposed alternate solutions don't fit the bill. 

If the nail is 10 cents, then the hammer is a dollar more: $\$1.10$, and then the total is $\$1.20$.

If the nail is 6 cents and the hammer is $\$1.04$, then that is not a dollar more for the hammer; it's 98 cents more.
Given that H+N = 1.1 and H = N+1 we can solve for N for both. H = 1.1-N AND H = 1+N so 1.1-N = 1+N, then 1.1 = 1+2N, so 0.1 = 2N, therefore .1/2 = N, thus .05 = N. Substitute .05 for N either original equation and H = 1.05.