#### Question

The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is Rs. 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is Rs. 90 whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is Rs. 70. Find the cost of each item per dozen by using matrices.

#### Solution

Let Rs.’x’, Rs.’y’ and Rs.’z’ be the cost of one dozen pencils, one dozen pens and one dozen erasers.

Thus, the system of equations are:

`{:(4x+3y+2z=60),(2x+4y+6z=90),(6x+2y+3z=70):}`

Let us write the above equations in the matrix form as:

`[[4,3,2],[2,4,6],[6,2,3]][[x],[y],[z]]=[[60],[90],[70]] " i.e "AX=B`

`"Using "R_z->R_2-1/2R_1 and R_3->R_3-3/2R_1`

`[[4,3,2],[0,5/2,5],[0,-5/2,0]][[x],[y],[z]]=[[60],[60],[-20]]`

`"Using "R_3->R_3+R_2`

`[[4,3,2],[0,5/2,5],[0,0,0]][[x],[y],[z]]=[[60],[60],[40]`

As matrix A is reduced to its upper triangular form we can write

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

5/2y + 5z = 60..........(ii)

5z = 40

z = 8.....(iii)

Substituting (iii) in (ii) we get,

5/2y + 5(8) = 60

y = 8.....(iv)

Substituting (iii) and (iv) in (i) we get,

4x + 3 (8) + 2 (8) = 60

x = 5

∴ Cost of one dozen pencils, one dozen pens and one dozen erasers is Rs. 5, Rs. 8 and Rs. 8 respectively