a. Express Equations (5) as a homogeneous linear system of three equations in four unknowns (x,...

   

 

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a. 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. b. Find the smallest solution for which all four variables are positive integers. ç Show that the solution given in the example is included among your solutions. • EXAMPLE 5 India (fourth century A.D.) The Bakhshali Manuscript is an ancient work of Indian/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 of linear 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 off in the value of their animals if each gives two animals, one to each of the others. Find the price of each animal and the total value of the 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 0w lellud] Aosid odT) eb lalod) bolageb wolley uri to otodw odr The method of solution described in the manuscript begins by subtracting the quai tity (x +y+ z) from both sides of the three equations to obtain 4x = 6y = 7z = K (x+y+ z). This shows that if the prices x, y, and z are to be integers, then the quanig K - (x +y+z) must be an integer that is divisible by 4, 6, and 7. The manuscript takes the product of these three numbers, or 168, for the value of K - (x +y+ " which yields x = (See Exercise 6 for more solutions to this problem.) to back substitution to oh 42, y = 28, and z = 24 for the prices and K 262 for the total value. %3D a. 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. b. Find the smallest solution for which all four variables are positive integers. ç Show that the solution given in the example is included among your solutions. • EXAMPLE 5 India (fourth century A.D.) The Bakhshali Manuscript is an ancient work of Indian/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 of linear 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 off in the value of their animals if each gives two animals, one to each of the others. Find the price of each animal and the total value of the 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 0w lellud] Aosid odT) eb lalod) bolageb wolley uri to otodw odr The method of solution described in the manuscript begins by subtracting the quai tity (x +y+ z) from both sides of the three equations to obtain 4x = 6y = 7z = K (x+y+ z). This shows that if the prices x, y, and z are to be integers, then the quanig K - (x +y+z) must be an integer that is divisible by 4, 6, and 7. The manuscript takes the product of these three numbers, or 168, for the value of K - (x +y+ " which yields x = (See Exercise 6 for more solutions to this problem.) to back substitution to oh 42, y = 28, and z = 24 for the prices and K 262 for the total value. %3D