Let the number of quarters be x
Then the number of nickels, using
ONE PART = TOTAL MINUS OTHER PART,
is 34-x.
                      Value      Value
Type       Number       of         of
 of          of        EACH       ALL
coin        coins      coin      coins
-------------------------------------------
quarters     x         $0.25     $0.25x
nickels   34-x         $0.05     $0.05(34-x)
-------------------------------------------
TOTALS      34        -----      $5.90

 The equation comes from the column on the right

  

0.25x + 0.05(34-x) = 5.90

Get rid of decimals by multiplying every term by 100

     25x + 5(34-x) = 590

    25x + 170 - 5x = 590

               20x = 420

                 x = 21

                 x = 21 = the number of quarters

The number of nickels is 34-x or 34-21 or 13 nickels

Checking:

21 quarters = 21($0.25) = $5.25
13 nickels  = 13($0.05) =   .65
-------------------------------
                  total = $5.90 

So 21 quarters and 13 nickels is the answer.

Edwin