Let the number of fives be x
Let the number of tens be y


                      Value      Value
Type       Number       of         of
 of          of        EACH       ALL
bill        bills      bill      bills
-------------------------------------------
fives        x          $5       $5x
tens         y         $10      $10y
-------------------------------------------
TOTALS      90        -----     $645

 The first equation comes from the second column.

  

                 x + y = 90

 The second equation comes from the last column.
  

           5x + 10y = 645

 So we have the system of equations:
           .

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

           x + y = 90
               y = 90 - x

Substitute (90 - x) for y in 5x + 10y = 645

   5x + 10(90 - x) = 645
    5x + 900 - 10x = 645
         -5x + 900 = 645
               -5x = -255
                 x = 51 = the number of fives.

Substitute in y = 90 - x
              y = 90 - (51)
              y = 39 tens.

Checking:  51 fives is $255 and 39 tens is $390
            That's 90 bills.
            And indeed $255 + $390 = $645
Edwin