Shannon needs 20g of 80% gold to make a pendant. She has some 85% gold and some 70% gold. How much of each should she use of each alloy, use two linear equations to solve

Two equations can be written to describe the desired relationships. One can describe the total quantity of gold alloy needed. The other can describe the amount of gold in the combination of alloys.

Let x and y represent the quantities in grams of 85% and 70% gold alloy needed, respectively.

x + y = 20 . . . . . . Shannon needs a total of 20 grams

0.85x +0.70y = 0.80(20) . . . . . the amount of gold in the mix of alloys

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These equations are often nicely solved by substitution. Substituting for y, we have ...

0.85x +0.70(20 -x) = 0.80(20)

0.15x = 0.10(20) . . . . . . . eliminate parentheses, subtract 0.70(20)

x = 40/3 = 13 1/3 . . . . . divide by 0.15

y = 20 -13 1/3 = 6 2/3

Shannon should use 13 1/3 grams of 85% alloy and 6 2/3 grams of 70% alloy.