Question

In a market survey three commodities A, B and C were considered. In finding out the index number some fixed weights were assigned to the three varieties in each of the commodities. The table below provides information regarding the consumption of three commodities according to the three varieties and also the total weight received by the commodity.

Commodity	Total weight			Total weight
	I	II	III
A	1	2	3	11
B	2	4	5	21
C	3	5	6	27

Find the weights assigned to the three varieties by using Cramer’s Rule

Answer 1

Let x, y and z be are consumption of three commodities A, B and C respectively.

Given that

x + 2y + 3z = 11  ........(1)

2x + 4y + 5z = 21  ........(2)

3x + 5y + 6z = 27  ........(2)

Here Δ = `|(1, 2, 3),(2, 4, 5),(3, 5, 6)|`

= 1(24 – 25) – 2(12 – 15) + 3(10 – 12)

= 1(1) 2(3) + 3(– 2)

= – 1 + 6 – 6

= – 1 ≠ 0

∴ We can apply Cramer’s Rule

Now `Delta_x = |(11, 2, 3),(21, 4, 5),(27, 5, 6)|`

= 11(24 – 25) – 2(126 – 135) + 3(105 – 108)

= 11(–1) – 2(– 9)+ 3(– 3)

= – 11 + 18 – 9

= – 2

`Delta_y = |(1, 11, 3),(2, 21, 5),(3, 27, 6)|`

= 1(126 – 135) – 11(12 – 15) + 3(54 – 63)

= 1(– 9) -11(– 3) + 3(– 9)

= – 9 + 33 – 27

= – 3

`Delta_z = |(1, 2, 11),(2, 4, 21),(3, 5, 27)|`

= 1(108 – 105) – 2(54 – 63) + 11(10 – 12)

= 1(3) – 2(– 9) + 11(– 2)

= 3 + 18 – 22

= – 1

∴ By Cramer’s Rule

x = `Delta_x/Delta = (-2)/(-1)` = 2

y = `Delta_y/Delta = (-3)/(-1)` = 3

z = `Delta_y/Delta = (-1)/(-1)` = 1

∴ (x, y, z) = (2, 3, 1)