Let the number of nickles be x
Let the number of dimes be y


                      Value      Value
Type       Number       of         of
 of          of        EACH       ALL
coin        coins      coin      coins
-------------------------------------------
nickles      x      $0.05       $0.05x
dimes        y      $0.10       $0.10y
-------------------------------------------
TOTALS      59      -----       $4.05

 The first equation comes from the number of coins column.

  

                   x + y = 59

 The second equation comes from the last column.
  

           0.05x + 0.10y = 4.05

Get rid of decimals by multiplying every term by 100:

                5x + 10y = 405

 So we have the system of equations:
           .

We solve by substitution.  Solve the first equation for y:

                   x + y = 59
                       y = 59 - x

Substitute (59 - x) for y in 5x + 10y = 405

         5x + 10(59 - x) = 405
          5x + 590 - 10x = 405
               -5x + 590 = 405
                     -5x = -185
                       x = 37 = the number of nickles.

Substitute in          y = 59 - x
                       y = 59 - (37)
                       y = 22 dimes.

Checking:  37 nickles is $1.85 and 22 dimes is $2.20
           That's 59 coins.
           And indeed $1.85 + $2.20 = $4.05

Edwin