• From an ancient Chinese book (Chiu-chang Suan-shu - 200 B.C.).

3 sheaves of a good crop, 2 sheaves of mediocre crop, and 1 sheaf of bad crop are sold for 39 dou.

2 sheaves of good, 3 mediocre, and 1 bad are sold for 34 dou.

1 good, 2 mediocre and 3 bad are sold for 26 dou.

What is the price received for each sheaf of a good crop, each sheaf of a mediocre crop, and each sheaf of a bad crop?

Today this problem can be formulated as 3 equations in 3 unknowns, where x, y, and z represent the price of a good, mediocre, and bad crop, respectively:

• I have Lemon Rum with an alcohol content of 30% and Rum with an alcohol content of 40%. I want to get 300 ml of cocktail with an alcoholic strength of 25% by adding equal quantities of Lemon Rum and Rum to orange juice. How much should I take from each of the three ingredients?

Let x, y, and z be the quantities (in ml) of Lemon Rum, Rum and orange juice.

x+y+z = 300 (300 ml of cocktail); x−y = 0 (equal quantities of Lemon Rum and Rum); 30%·x + 40%·y = 25%·300
(Lemon Rum with 30% of alcohl, Rum with 40% of alcohl → 300 ml with 25% of alcohl).

107 ml of Lemon Rum and Rum, 86 ml of orange juice.

• I have three bars made up of silver, copper and tin. Their compositions are:

silver copper tin 1^ 80% 17.5% 2.5% 2^ 70% 25% 5% 3^ 60% 30% 10%

How much metal does it take from each bar to get a metal composed in the following way?

silver: 74%, copper: 21.625%, tin: 4.375%

Let x, y and z be the fractions that must be taken from each bar.