Question

The cost of 2kg. of wheat and 1kg. of sugar is ₹ 100. The cost of 1kg. of wheat and 1kg. of rice is ₹ 80. The cost of 3kg. of wheat, 2kg. of sugar and 1kg of rice is ₹ 220. Find the cost of each per kg., using Cramer’s rule

Answer 1

Let cost of 1 kg of wheat be x; cost of 1kg of sugar be y and cost of 1 kg of rice be z.

2x + y = 100

x + z = 80

3x + 2y + z = 220

Here `Delta = |(2, 1, 0),(1, 0, 1),(3, 2, 1)|`

= 2(0 – 2) – 1(1 – 3)

= – 4 + 2

= – 2 ≠ 0

∴ We can apply Cramer’s Rule and the system is consistent and it has unique solution.

`Delta_x = |(100, 1, 0),(80, 0, 1),(220, 2, 1)|`

= 100(0 – 2) – 1(80 – 220)

= – 200 – 1(– 140)

= – 200 +140

= – 60

`Delta_y = |(2, 100, 0),(1, 80, 0),(3, 220, 1)|`

= 2(80 – 220) -100(1 – 3)

= 2(–140) – 100(– 2)

= – 280 + 200

= – 80

`Delta_x = |(2, 1, 100),(1, 0, 80),(3, 2, 220)|`

= 2(0 – 160) – 1(220 – 240) + 100(2 – 0)

= 2(– 160) – 1(– 20) + 100(2)

= – 320 + 20 + 200

= – 100

∴ By cramer’s rule

x = `Delta_x/Delta = (- 60)/(-2)` = 30

y = `Delta_y/Delta = (-80)/(-2)` = 40

z = `Delta_z/Delta = (- 100)/(- 2)` = 50

The solution is (x, y, z) = (30, 40, 50)

∴ The cost of 1 kg of wheat is Rs 30; the cost of 1 kg of sugar is ₹ 40 and the cost of 1 kg of rice is ₹ 50.