Let the number of dimes be x
Then the number of nickels, using
ONE PART = TOTAL MINUS OTHER PART,
is 52-x.
                      Value      Value
Type       Number       of         of
 of          of        EACH       ALL
coin        coins      coin      coins
-------------------------------------------
dimes         x       $0.10     $0.10x
nickels    52-x       $0.05     $0.05(52-x)
-------------------------------------------
TOTALS      52        -----     $3.55

 The equation comes from the column on the right

  

0.10x + 0.05(52-x) = 3.55

Get rid of decimals by multiplying every term by 100:

     10x + 5(52-x) = 355

    10x + 260 - 5x = 355

          5x + 260 = 355

                5x = 95

                 x = 19 = the number of dimes.

The number of nickels is 52-x or 52-19 or 33 nickels.

Checking:  19 dimes is $1.90 and 33 nickels is $1.65
            That's 52 coins.
            And indeed $1.90 + $1.65 = $3.55
Edwin