SOLUTION: Three are three type ingots. The first has 20gms gold, 30gms of silver and 40 gms of copper. the second has 30gms of gold, 40gms of silver and 50 gms copper. The third has 40 gms

Algebra ->  Systems-of-equations -> SOLUTION: Three are three type ingots. The first has 20gms gold, 30gms of silver and 40 gms of copper. the second has 30gms of gold, 40gms of silver and 50 gms copper. The third has 40 gms       Log On


   

Question 844459: Three are three type ingots. The first has 20gms gold, 30gms of silver and 40 gms of copper. the second has 30gms of gold, 40gms of silver and 50 gms copper. The third has 40 gms of gold, 50gms silver and 90 gms copper. What weight will be taken from each of the previous ingots to form a new ingot of 34 gms gold, 46 gms silver and 67 gms copper?
Answer by josgarithmetic(39616) About Me  (Show Source):
You can put this solution on YOUR website!
x = amount from first 90 gram ingot
y = amount from second 120 gram ingot
z = amount from third 180 gram ingot

Account for gold: %282%2F9%29x%2B%283%2F12%29y%2B%284%2F18%29z=34
Account for silver: %283%2F9%29x%2B%284%2F12%29y%2B%285%2F18%29z=46
Account for copper: %284%2F9%29x%2B%285%2F12%29y%2B%289%2F18%29z=67

Simplify the system first, to have lowest-term fraction coefficients:
Gold: %282%2F9%29x%2B%281%2F4%29y%2B%282%2F9%29z=34
Silver: %281%2F3%29x%2B%281%2F3%29y%2B%285%2F18%29z=46
Copper: %284%2F9%29x%2B%285%2F12%29y%2B%281%2F2%29z=67

Simplify the system next multiplying each equation by its lowest common denominator to clear the fractions; using 36 for gold and copper, and using 18 for the silver:
Gold: 8x%2B9y%2B8z=34%2A36
Silver: 6x%2B6y%2B5z=46%2A18
Copper: 16x%2B15y%2B18z=67%2A36

Finish the constant term multiplications:
Gold: 8x%2B9y%2B8z=1224
Silver: 6x%2B6y%2B5z=828
Copper: 16x%2B15y%2B18z=2412

A software tool for matrices would be best way unless you really wanted to do all the matrix row operations on paper. A useful choice is http://www.math.purdue.edu/~dvb/matrix.html
'
-----------------------------------
RESULT:
(x, y, z) is (45, 48, 54)
MEANING: 45 of number 1 ingots, 48 of number 2 ingots, 54 of number 3 ingot.
-----------------------------------