Let the number of nickels be x
Then the number of quarters, using

ONE PART = TOTAL MINUS OTHER PART,

is 23-x.
                      Value      Value
Type       Number       of         of
 of          of        EACH       ALL
coin        coins      coin      coins
-------------------------------------------
NICKELS      x        $0.05    $0.05x
QUARTERS   23-x       $0.10    $0.10(23-x)
-------------------------------------------
TOTALS      23        -----    $2.35

 The equation comes from the column on the right

  

0.05x + 0.25(23-x) = 2.35

Get rid of decimals by multiplying every term by 100

     5x + 25(23-x) = 235

    5x + 575 - 25x = 235

        -20x + 575 = 235

              -20x = -340

                 x = 17 = the number of nickels

The number of quarters is 23-x or 23-17 or 6 quarters.

Checking:  17 nickels is 17($0.05) = $0.85 
and 6 quarters is 6($.25) = %1.50

Indeed that's 17+6 = 23 coins and $0.85+$1.50 = $2.35.

Edwin