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The Bakhshali Manuscript is an ancient work ofIndian/Hindu mathematics dating from around the fourth century A.D., although some of its materials undoubtedly come from many centuries before. It consists of about 70 leaves or sheets of birch bark containing mathematical problems and their solutions. Many of its problems are so-called equal-ization problems that lead to systems oflinear equations. One such problem on the frag-ment shown is the following:
One merchant has seven asava horses, a second has nine haya horses, and a third has ten camels. They are equally well offin the value of their animals if each gives two animals, one to each ofthe others. Find the price of each animal andthe total value ofthe animals possessed by each merchant.
Let x be the price of an asava horse, let y be the price of a haya horse, let z be the price of a camel, and the let K be the total value of the animals possessed by each merchant. Then the conditions of the problem lead to the following system of equations:
5x + y + z = K
x + 7y + z = K
x + y + 8z = K
- Express Equations (5) as a homogeneous linear system of three equations in four unknowns (x, y, z, and K), and show that the solution set has one arbitrary parameter.
- Find the smallest solution for which all four variables are positive integers.
- Show that the solution given in the example is included among your solutions
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