SOLUTION: Dave Bowers collects U.S. gold coins. He has a collection of 47 coins. Some are​ $10 coins, and the rest are​ $20 coins. If the face value of the coins is ​$700​, how many

Let x be the number of the $20 coins.

Then the number of $20 coins is (47-x).


Then the "face value equation" is


    20x + 10*(47-x) = 700   dollars.


From the equation, express x and calculate


    x =  = 23.


ANSWER.  23 coins are the $20 coins and the rest 47-23 = 24  are the $10 coins. 


CHECK.   23*20 + 24*10 = 700 dollars.   ! Correct !

Let x be the number of the $20 coins, and

let y be the number of $10 coins.


From the condition, you have these two equations


    x  +   y =  47   coins       (2)

   20x + 10y = 700   dollars.    (3)


From the first equation, express  y = 47-x  and substitute it into the second equation.


You will get an equation for single unknown


    20x + 10*(47-x) = 700.      (4)


Equation (4) is identical to equation (1).
Therefore, the rest of solution is the same as in the first method, giving (of course) the same answer.