Let the number of dimes be x.
Then the number of quarters, 
using ONE PART = TOTAL MINUS OTHER PART,
  is 50-x. 

Make this chart:

                      Value      Value
Type       Number       of         of
 of          of        EACH       ALL
coin        coins      coin      coins
-------------------------------------------
DIMES         x        $0.10    $0.10x 
QUARTERS    50-x       $0.25    $0.25(50-x)
-------------------------------------------
TOTALS       50        -----    $8.60

 

 0.10x + 0.25(50-x) = 8.60
Get rid of decimals by multiplying every term by 100
     10x + 25(50-x) = 860   
   10x + 1250 - 25x = 860
        -15x + 1250 = 860
              -15x = -390
                 x = 26 = the number of dimes
The number of quarters is 50-x or 50-26 or 24 quarters.

Checking: 26 dimes + 24 quarters = 50 coins. That checks.
26 dimes is worth $2.60 and 24 quarters is worth $6.00.
That adds to $8.60.  That checks.  So we are right.

Edwin