Question 1033124
A certain quantity was taken from a ten liter 30% alcohol 
solution and replaced by pure alcohol so as to produce 51% 
alcohol solution. How many liters were taken and replaced?
<pre>
Let the number of liters of pure added be x
Let the number of liters of 30%alcohol left
    in the jug after the x liters were 
    taken out and replaced be 
    y liters of pure alcohol

                         Percent      Liters
                         alcohol     of pure
  Type        Liters       in        alcohol
   of          of         EACH      contained
liquid       liquid    as decimal    in each
---------------------------------------------
pure alcohol    x         1.00       1.00x
30% alcohol     y         0.30       0.30y
----------------------------------------------------
Final mixture  10         0.51      (0.51)(10) = 5.1

 The first equation comes from the second column.

  {{{(matrix(4,1,Liters,of,liquid,added))}}}{{{""+""}}}{{{(matrix(6,1,Liters,of,liquid,left,in,jug))}}}{{{""=""}}}{{{(matrix(4,1,Liters,in,full,jug))}}}

                 x + y = 10

 The second equation comes from the last column.
  {{{(matrix(5,1,Liters,of,pure,alcohol,added))}}}{{{""+""}}}{{{(matrix(10,1,Liters,of,pure,alcohol,contained,in,liquid,left,in,jug ))}}}{{{""=""}}}{{{(matrix(10,1,Liters,of,pure,alcohol,contained,in,jug,in,the,end))}}}

           1.00x + 0.30y = 5.1

Get rid of decimals by multiplying every term by 10:

          10x + 3y = 51

 So we have the system of equations:
           {{{system(x + y = 10,10x + 3y = 51)}}}.

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

           x + y = 10
               y = 10 - x

Substitute (10 - x) for y in 10x + 3y = 51

     10x + 3(10 - x) = 51
       10x + 30 - 3x = 51
             7x + 30 = 51
                  7x = 21
answer:            x = 3 liters drained and 
                         replaced with pure alcohol

Edwin</pre>