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


                      Value      Value
Type       Number       of         of
 of          of        EACH       ALL
coin        coins      coin      coins
-------------------------------------------
nickels       x      $0.05       $0.05x
dimes         y      $0.10       $0.10y
-------------------------------------------
TOTALS       45      -----       $3.65

The first equation comes from the "Number of coins" column.

  

                   x + y = 45

 The second equation comes from the last column.
  

           0.05x + 0.10y = 3.65

Get rid of decimals by multiplying every term by 100:

                5x + 10y = 365

 So we have the system of equations:
               .

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

                    x + y = 45
                        y = 45 - x

Substitute (45 - x) for y in 5x + 10y = 365

          5x + 10(45 - x) = 365
           5x + 450 - 10x = 365
                -5x + 450 = 365
                      -5x = -85
                        x = 17 = the number of nickels.

          Substitute in y = 45 - x
                        y = 45 - (17)
                        y = 28 dimes.

Checking:  17 nickels is $0.85 and 28 dimes is $2.80
            That's 45 coins.
            And indeed $0.85 + $2.80 = $3.65
Edwin