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Question
The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is ₹ 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is ₹ 90. Whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is ₹ 70. Find the cost of each item per dozen by using matrices
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Solution
Let the cost of 1 dozen pencils, 1 dozen pens and 1 dozen erasers be ₹ x, ₹ y and ₹ z respectively.
According to the given conditions,
4x + 3y + 2z = 60
2x + 4y + 6z = 90
i.e. x + 2y + 3z = 45
6x + 2y + 3z = 70
Matrix form of the given system of equations is,
`[(4, 3, 2),(1, 2, 3),(6, 2, 3)] [(x),(y),(z)] = [(60),(45),(70)]`
Applying R1 ↔ R2
`[(1, 2, 3),(4, 3, 2),(6, 2, 3)] [(x),(y),(z)] = [(45),(60),(70)]`
Applying R2 → R2 − 4R1, R3 → R3 − 6R1
`[(1, 2, 3),(0, -5, -10),(0, -10, -15)] [(x),(y),(z)] = [(45),(-120),(-200)]`
Applying R3 → R3 − 2R2,
`[(1, 2, 3),(0, -5, -10),(0, 0, 5)] [(x),(y),(z)] = [(45),(-120),(40)]`
Hence, the original matrix is reduced to an upper triangular matrix.
∴ By equality of matrices, we get
x + 2y + 3z = 45 .......(i)
−5y − 10z = −120
i.e. y + 2z = 24 .......(ii)
5z = 40 .......(iii)
i.e. z = 8
Substituting z = 8 in equation (ii), we get
y + 2(8) = 24
∴ y = 8
Substituting z = 8 and y = 8 in equation (i), we get
x + 2(8) + 3(8) = 45
∴ x + 16 + 24 = 45
∴ x = 5
∴ x = 5, y = 8, z = 8
Thus, the cost of pencils is ₹ 5 per dozen, that of pens is ₹ 8 per dozen and that of erasers is ₹ 8 per dozen.
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