Answer to Question #322472 in Linear Algebra for Kenth Smith

Question

How much copper and how much magnesium should be added to 200lbs of an alloy containing 20% copper and 40% magnesium to produce an alloy containing 24% copper and 50% magnesium?

Answer 1

200lbs of an alloy containing 20% copper and 40% magnesium contains 0.2*200 = 40lbs of copper, 0.4*200 = 80lbs of magnesium and 200 - 40 - 80 = 80lbs others metals.

If we add x lbs of copper there will be x+40 lbs of copper in the new alloy.

If we add y lbs of magnesium there will be y+80 lbs of magnesium in the new alloy.

The total weight of the new alloy will be x + y + 200lbs, and new concentration of copper (x+40) / (x+y+200) = 0.24.

The new concentration of magnesium is (y+80) / (x+y+200) = 0.5.

Multiplying both equations by x+y+200 get system of two linear equations:

X + 40 = 0.24*x + 0.24*y + 48

y + 80 = 0.5*x + 0.5*y + 100

 76*x - 24*y = 800

-50*x + 50*y = 2000

Solving this system get x = 440/13 and y = 960/13