This problem has a typo.  I think the number of coins should be 37, 
certainly not 17. I'll first do it with 17 to show you that there can be 
no way to have $6.70 in quarters and dimes with only 17 coins.  If 
you think about it, you know that's not possible because if they were all
quarters, 17 quarters is only $4.50

Then below that I'll do it changing the 17 to 37.

Let the number of quarters be x
Then the number of dimes, using
ONE PART = TOTAL MINUS OTHER PART,
is 17-x.
                      Value      Value
Type       Number       of         of
 of          of        EACH       ALL
coin        coins      coin      coins
-------------------------------------------
quarters      x        $0.25    $0.25x
dimes      17-x        $0.10    $0.25(17-x)
-------------------------------------------
TOTALS       17       -----     $6.70

 The equation comes from the column on the right

  

 0.25x + 0.10(17-x) = 6.70

Get rid of decimals by multiplying every term by 100

     25x + 10(17-x) = 670

    25x + 170 - 10x = 670

          15x + 170 = 670

               15x = 500

                 x = 33.3333333 = the number of quarters 

The number of dimes is 17-x or 17-33.3333333 or -16.3333333 dimes

You cannot have a fraction number of quarters and a negative number
of dimes.  There is no solution.


-------------------------------------------------------

Now here's the same problem using 37 coins instead of 17 coins.


Let the number of quarter be x
Then the number of dime, using
ONE PART = TOTAL MINUS OTHER PART,
is 37-x.
                      Value      Value
Type       Number       of         of
 of          of        EACH       ALL
coin        coins      coin      coins
-------------------------------------------
quarter      x         $0.25     $0.25x
dime      37-x         $0.1      $0.1(37-x)
-------------------------------------------
TOTALS      37        -----      $6.70

 The equation comes from the column on the right

  

0.25x + 0.10(37-x) = 6.70

Get rid of decimals by multiplying every term by 100

     25x + 10(37-x) = 670

    25x + 370 - 10x = 670

          15x + 370 = 670

               15x = 300

                 x = 20 = the number of quarters

The number of dimes is 37-x or 37-20 or 17 dimes

Edwin