Question

(i) The cost of 5 kg of sugar, 5kg of wheat and 2kg of rice is Rs. 340. The cost of 4 kg of rice, 2kg of sugar and 5kg of wheat is Rs. 270. Rs. 264 is the cost of 3kg of wheat, 2kg of rice and 4 kg of sugar. Represent the above problem by a system of linear equations and find the cost of each item per kg by matix method. (ii) How increase in the prices of essential commodities affect economic weaker section of society?

Answer 1

Answer:

Sugar = Rs 40, Wheat = Rs 18 and Rice = Rs 25

Step-by-step explanation:

Let the cost of 1 kg of sugar be Rs x

Let the cost of 1 kg of wheat be Rs y

Let the cost of 1 kg of rice be Rs z

∴Cost of 5 kg sugar, 5 kg wheat, 2 kg rice = Rs 340

 Cost of 2 kg sugar, 5 kg wheat, 4 kg rice = Rs 270

 Cost of 4 kg sugar, 3 kg wheat, 2 kg rice = Rs 264

Required system of linear equations:

5x + 5y + 2z = 340

2x + 5y + 4z = 270

4x + 3y + 2z = 264  

Let A = $\left[\begin{array}{ccc}5&5&2\\2&5&4\\4&3&2\end{array}\right]$ , X = $\left[\begin{array}{ccc}x\\y\\z\end{array}\right]$ and B =$\left[\begin{array}{ccc}340\\270\\264\end{array}\right]$

∴ AX = B

Pre multiplying by A⁻¹,

X =  A⁻¹B  ..............1

For matrix A,

|A| = 5(10-12) -5(4-16) +2(6-20)

or |A| = 22

Thus, A is invertible

Finding cofactors of matrix A

C₁₁ = -2, C₁₂ = 12, C₁₃ = -14

C₂₁ = -4, C₂₂ = 2, C₂₃ = 5

C₃₁ = 10, C₃₂ = -16, C₃₃ = 15

∴Cofactor matrix of A = $\left[\begin{array}{ccc}-2&12&-14\\-4&2&5\\10&-16&15\end{array}\right]$

∴ adj A = $\left[\begin{array}{ccc}-2&-4&10\\12&2&-16\\-14&5&15\end{array}\right]$

Therefore,  A⁻¹ = 1/ |A| ( adj A)

or A⁻¹ = 1/22 $\left[\begin{array}{ccc}-2&-4&10\\12&2&-16\\-14&5&15\end{array}\right]$

Substituting in equation 1,

$\left[\begin{array}{ccc}x\\y\\z\end{array}\right]$  = 1/22 $\left[\begin{array}{ccc}-2&-4&10\\12&2&-16\\-14&5&15\end{array}\right]$ $\left[\begin{array}{ccc}340\\270\\264\end{array}\right]$

or  $\left[\begin{array}{ccc}x\\y\\z\end{array}\right]$ = 1/22$\left[\begin{array}{ccc}-680-1080+2640\\4080+540-4224\\-4760+1350+3960\end{array}\right]$

or $\left[\begin{array}{ccc}x\\y\\z\end{array}\right]$ = 1/22$\left[\begin{array}{ccc}880\\396\\550\end{array}\right]$

or $\left[\begin{array}{ccc}x\\y\\z\end{array}\right]$ = $\left[\begin{array}{ccc}40\\18\\25\end{array}\right]$

Thus, x=40, y=18, z=25

Cost of 1 kg of sugar is Rs 40

Cost of 1 kg of wheat is Rs 18

Cost of 1 kg of rice is Rs 25