Let the number of quarters be x
Let the number of dollars be y


                      Value      Value
Type       Number       of         of
 of          of        EACH       ALL
coin        coins      coin      coins
-------------------------------------------
quarters       x      $0.25     $0.25x
dollars        y      $1.00     $1.00y
-------------------------------------------
TOTALS        45      -----     $38.25

 The first equation comes from the second column.

  
                 x + y = 45

 The second equation comes from the last column.
  

         0.25x + 1.00y = 38.25

Get rid of decimals by multiplying every term by 100:

            25x + 100y = 3825

 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 25x + 100y = 3825

    25x + 100(45 - x) = 3825
    25x + 4500 - 100x = 3825
          -75x + 4500 = 3825
                 -75x = -675
                    x = 9 = the number of quarters.

Substitute in y = 45 - x
              y = 45 - (9)
              y = 36 dollars.


Checking:  9 quarters is $2.25 and 36 dollars is $36.00
            That's 45 coins.
            And indeed $2.25 + $36.00 = $38.25
Edwin