The federal tax was $500 more than the state tax. if the sum of the taxes was $6900, what was the amount of federal tax?

So Federal Tax (F) is $500 more than the State Tax (S).  
This would give us the equation F = S + 500.  
If F + S = 6900, we now have two equations and can substitute.
Because F = S +500, we can substitute S + 500 for F in the second equation. 
This gives us (S+500) + S = 6900.
This simplifies to S+S=6900-500 or 2S = 6400.
This would make S = 3200.  
Plugging 3200 in for S in the second equation yields F = 3700.
So $3,700 is the Federal Tax.