Question 844459
x = amount from first  90 gram ingot
y = amount from second 120 gram ingot
z = amount from third 180 gram ingot


Account for gold: {{{(2/9)x+(3/12)y+(4/18)z=34}}}
Account for silver: {{{(3/9)x+(4/12)y+(5/18)z=46}}}
Account for copper: {{{(4/9)x+(5/12)y+(9/18)z=67}}}


Simplify the system first, to have lowest-term fraction coefficients:
Gold:  {{{(2/9)x+(1/4)y+(2/9)z=34}}}
Silver: {{{(1/3)x+(1/3)y+(5/18)z=46}}}
Copper: {{{(4/9)x+(5/12)y+(1/2)z=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+9y+8z=34*36}}}
Silver: {{{6x+6y+5z=46*18}}}
Copper: {{{16x+15y+18z=67*36}}}


Finish the constant term multiplications:
Gold: {{{8x+9y+8z=1224}}}
Silver: {{{6x+6y+5z=828}}}
Copper: {{{16x+15y+18z=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

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RESULT:
(x, y, z) is (45, 48, 54)
MEANING:  45 of number 1 ingots, 48 of number 2 ingots, 54 of number 3 ingot.
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