Question

The price of three commodities, X, Y and Z are and z respectively Mr. Anand purchases 6 units of Z and sells 2 units of X and 3 units of Y. Mr.Amar purchases a unit of Y and sells 3 units of X and 2 units of Z. Mr. Amit purchases a unit of X and sells 3 units of Y and a unit of Z. In the process they earn ₹ 5,000/-, ₹ 2,000/- and ₹ 5,500/- respectively. Find the prices per unit of three commodities by rank method

Answer 1

Let the equations for Mr. Anand, Mr. Amar, and Mr. Amit are

2x + 3y – 6z = 5000

3x – y + 2z = 2000

– x + 3y + z = 5500 respectively

The matrix equation corresponding to the given system is

`[(2, 3, -6),(3, -1, 2),(-1, 3, 1)] [(x),(y),(y)] = [(5000),(2000),(5500)]`
             A                X    =       B

Augmented Matrix [A, B]	Elementary Tranformation
`[(2, 3, -6, 5000),(3, -1, 2, 2000),(-1, 3, 1, 5500)]`
`∼[(-1, 4, -8, 3000),(3, -1, 2, 2000),(-1, 3, 1, 5500)]`	`{:"R"_1 -> "R"_1 - "R"_2:}`
`∼[(-1, 4, -8, 3000),(0, 11, -22, 11000),(-1, 3, 1, 5500)]`	`{:"R"_2 -> "R"_2 + 3"R"_1:}`
`∼[(-1, 4, -8, 3000),(0, 1, -2, 1000),(-1, 3, 1, 500)]`	`{:"R"_2 -> "R"_2/11:}`
`∼[(-1, 4, -8, 3000),(0, 1, -2, 1000),(0, -1, 9, 500)]`	`{:"R"_3 -> "R"_3 - "R"_1:}`
`∼[(-1, 4, -8, 3000),(0, 1, -2, 1000),(0,0, 7, 3500)]`	`{:"R"_3 -> "R"_3 + "R"_2:}`
p(A) = 3; p(A, B) = 3

∴ The given system is equivalent to the matrix equation

`[(-1, 4, -8),(0, 1, -2),(0, 0, 7)] [(x),(y),(z)] = [(3000),(1000),(3500)]`

– x + 4y – 8z = 3000  ........(1)

y – 2z = 1000  ........(2)

7z = 3500  .........(3)

Equation (3) ⇒ z = `3500/7`

∴ z = 500

Eqn (2) ⇒ y – 2(500) = 1000

y = 1000 + 1000

∴ y = 2000

Eqn (1) ⇒ – x + 4(2000) – 8(500) = 3000

– x + 8000 – 4000 = 3000

– x + 4000 = 3000

– x = 3000 – 4000

– x = – 1000

∴ x = 1000

The price of three commodities, x, y and z are ₹ 1000, ₹ 2000 and ₹ 500 respectively.