SOLUTION: An alloy contains 20% silver and 30% lead. How much silver and how much lead should be added to 100 pounds of the alloy in order to obtain 25% silver and 33% lead

The initial amount of the alloy is 100 pounds.

Let x = the amount (the mass) of silver added, and

let y be the amount of lead added.


Then the total mass of the alloy will be  (100 + x + y) pounds


From the condition, you have these two equations


     = 0.25   (25% of silver in the new alloy)

     = 0.33   (33% of lead in the new alloy)


Equivalently


    20 + x = 0.25*(100 + x + y)    (1)

    30 + y = 0.33(*100 + x + y)    (2)


Simplify to get


    0.75x - 0.25y = 5

   -0.33x + 0.67y = 3


Apply the determinant method ( = Cramer's rule). The determinant of the coefficient matrix is


    det  = 0.75*0.67 -0.25*0.33 = 0.42.


The determinant of the  x-associated matrix is

    det  = 5*0.67 + 0.25*3 = 4.1.



The determinant of the  y-associated matrix is

    det  = 0.75*3 + 5*0.33 = 3.9.


Thus  x =  = 9.7619,  or  9.7619 pounds of silver.

      y =  = 9.2857,  or  9.2857 pounds of lead.


Answer.  9.7619  pounds of silver  and   9.2857  pounds of lead should be added.