Awoman bought 100 christmas cards. for the ones that sing a song when you open them, she paid 30 cents each. for the rest she paid 5 cents each. if the cards cost $10.25 in all, how many of the expensive kind did she buy?

Let the number of sing-a-songs be x
Let the number of ordinarys be y

                      Value      Value
Type       Number       of         of
of          of        EACH       ALL
card       cards       card      cards

sing-a-songs  x      $0.30       $0.30x
ordinarys     y      $0.05       $0.05y

TOTALS       100            $10.25

The first equation comes from the second column.

 

                 x + y = 100

The second equation comes from the last column.
 

           0.3x + 0.05y = 10.25

Get rid of decimals by multiplying every term by 100:

          30x + 5y = 1025

So we have the system of equations:
           .

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

           x + y = 100
               y = 100 - x

Substitute (100 - x) for y in 30x + 5y = 1025

  30x + 5(100 - x) = 1025
    30x + 500 - 5x = 1025
          25x + 500 = 1025
               25x = 525
                x = 21 = the number of sing-a-song cards.

Substitute in y = 100 - x
              y = 100 - (21)
              y = 79 ordinarys.

Checking:  21 sing-a-songs is $6.30 and 79 ordinarys is $3.95
            That's 100 cards.
            And indeed $6.30 + $3.95 = $10.25