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प्रश्न
The cost of 4 pencils, 3 pens, and 2 books is ₹ 150. The cost of 1 pencil, 2 pens, and 3 books is ₹ 125. The cost of 6 pencils, 2 pens, and 3 books is ₹ 175. Find the cost of each item by using matrices.
उत्तर
Let the cost of 1 pencil, 1 pen and 1 book be ₹ x, ₹ y, ₹ z respectively.
According to the given conditions,
4x + 3y + 2z = 150
x + 2y + 3z = 125
6x + 2y + 3z = 175
The equations can be written in matrix form as:
`[(4,3,2),(1,2,3),(6,2,3)] [("x"),("y"),("z")] = [(150),(125),(175)]`
By R1 ↔ R2, we get,
`[(1,2,3),(4,3,2),(6,2,3)] [("x"),("y"),("z")] = [(125),(150),(175)]`
By R2 - 4R1 and R3 - 6R1, we get,
`[(1,2,3),(0,-5,-10),(0,-10,-15)] [("x"),("y"),("z")] = [(125),(-350),(-575)]`
By R3 - 2R2, we get,
`[(1,2,3),(0,-5,-10),(0,0,5)] [("x"),("y"),("z")] = [(125),(-350),(125)]`
∴ `[("x" + 2"y" + 3"z"),(0 - "5y" - 10"z"),(0 + 0 + "5z")] = [(125),(-350),(125)]`
By equality of matrices,
x + 2y + 3z = 125 ...(1)
- 5y - 10z = - 350 ...(2)
5z = 125 ...(3)
From (3), z = 25
Substituting z = 25 in (2), we get
- 5y - 10(25) = - 350
∴ - 5y = - 350 + 250 = - 100
∴ y = 20
Substituting y = 20, z = 25 in (1), we get
x + 2(20) + 3(25) = 125
∴ x = 125 - 40 - 75 = 10
∴ x = 10, y = 20, y = 25
Hence, the cost of 1 pencil is ₹ 10, 1 pen is ₹ 20 and 1 book is ₹ 25.
Notes
[Note: Answer to cost of a pen in the textbook is incorrect.]
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