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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.
Solution
Let Rs.’x’, Rs.’y’ and Rs.’z’ be the cost of one pencil, one pen and one eraser.
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],[0,0,5]][[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.
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