Question

Complete the following activity.

The cost of 4 kg potato, 3kg wheat and 2kg rice is ₹ 60. The cost of 1 kg potato, 2 kg wheat and 3kg rice is ₹ 45. The cost of 6 kg potato, 3 kg rice and 2 kg wheat is ₹ 70. Find the per kg cost of each item by matrix method.

Solution: Let the cost of potato, wheat and rice per kg be x, y and z respectively.

Therefore by given conditions,

4x + ( )y + 2( ) = ( )

x + 2y + ( )( ) = ( )

( )x + 2y + 3z = ( )

Matrix form of above equations is,

`[("( )", 3, "( )"),(1, "( )", 3),("( )", 2, "( )")] [(x),(y),(z)] =[("( )"), (45), ("( )")]`

R₁ ↔ R₂

`[(1, 2, 3),("( )", "( )", "( )"),(6, 2, 3)] [(x),(y),(z)] =[("( )"), (60), ("( )")]`

R₂ – 4R₁, R₃ – 6R₁

`[(1, 2, 3),("( )", -5, "( )"),(0, "( )", -15)] [(x),(y),(z)] =[(45), ("( )"), (-200)]`

`(-1)/5 "R"_2, (-1)/5 "R"_3`

`[("( )", 2, 3),(0, "( )", 2),(0, 2, "( )")] [(x),("( )"),(z)] =[(45), (24), (40)]`

R₃ – 2R₂

`[(1, 2, 3),(0, 1, 2),(0, 0, -1)] [(x),(y),(z)] =[("( )"), ("( )"), ("( )")]`

By pre multiplying we get,

x + 2y + ( )z = ( )    .....(i)

y + 2z = 24    ......(ii)

–z = ( )      ......(iii)

From (iii), we get, z = ( )

From (ii), we get, y = ( )

From (i), we get, x = ( )

Therefore the cost of Potato, Wheat and Rice per kg are _______, _______ and _______ respectively.

Answer 1

Let the cost of potato, wheat and rice per kg be x, y and z respectively.

Therefore by given conditions,

4x + (3)y + 2(z) = (60)

x + 2y + (3)(z) = (45)

(6)x + 2y + 3z = (70)

Matrix form of above equations is,

`[(4, 3, 2),(1, 2, 3),(6, 2, 3)] [(x),(y),(z)] =[(60), (45), (70)]`

Applying R₁ ↔ R₂, we get

`[(1, 2, 3),(4, 3, 2),(6, 2, 3)] [(x),(y),(z)] =[(45), (60), (70)]`

Applying R₂ → R₂ – 4R₁, R₃ → R₃ – 6R₁, we get

`[(1, 2, 3),(0, -5, -10),(0, -10, -15)] [(x),(y),(z)] =[(45), (-120), (-200)]`

Applying R₂ → `(-1)/5 "R"_2`, R₃ → `(-1)/5 "R"_3`, we get

`[(1, 2, 3),(0, 1, 2),(0, 2, 3)] [(x),(y),(z)] =[(45), (24), (40)]`

Applying R₃ → R₃ – 2R₂, we get

`[(1, 2, 3),(0, 1, 2),(0, 0, -1)] [(x),(y),(z)]` = `[(45), (24), (-8)]`

By pre multiplying we get,

x + 2y + (3)z = (45)    .....(i)

y + 2z = 24    ......(ii)

– z = (– 8)      ......(iii)

From (iii), we get, z = 8

From (ii), we get

y + 2(8) = 24

∴ y = 24 – 16 = 8 

From (i), we get,

x + 2(8) + 3(8) = 45

∴ x + 16 + 24 = 45

∴ x = 45 – 40 = 5

Therefore the cost of Potato, Wheat and Rice per kg are 5, 8 and 8 respectively.