Question

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

Answer 1

Let Rs.‘x’, Rs.‘y’ and Rs.‘z’ be the cost of one pencil, one pen and one eraser.

Thus, the system of equations is

`{:(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],[0,0,5]][[x],[y],[z]] = [(60),(60),(40)]`

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

4x + 3y + 2z = 60     .....(i)

`5/2`y + 5z = 60     .....(ii)

0x + 0y + 5z = 40

z = 8     .....(iii)

Substituting (iii) in (ii), we get,

`5/2`y + 5(8) = 60

y = `(60 - 40)/5 xx 2 = 8`

y = 8     .....(iv)

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

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

4x = 60 − 24 − 16

x = `20/4 = 5`

∴ x = 5

∴ The cost of one pencil, one pen and one eraser is Rs. 5, Rs. 8 and Rs. 8 respectively.