Let the number of dimes be x
Then the number of nickels, using
ONE PART = TOTAL MINUS OTHER PART,
is 14-x.
                      Value      Value
Type       Number       of         of
 of          of        EACH       ALL
coin        coins      coin      coins
-------------------------------------------
dimes       x          $0.10   $0.10x
nickels   14-x         $0.05   $0.05(14-x)
-------------------------------------------
TOTALS      14        -----    $1.00

 The equation comes from the column on the right

  

   0.10x + 0.05(14-x) = 10

Get rid of decimals by multiplying every term by 100:

        10x + 5(14-x) = 100

        10x + 70 - 5x = 100

              5x + 70 = 100

                   5x = 30

                    x = 6 = the number of dimes.

The number of nickels is 14-x or 14-6 or 8 nickels.

Checking:  6 dimes is $0.60 and 8 nickels is $0.40

            And indeed $0.60 + $0.40 = $1.00
Edwin