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प्रश्न
Express the following equations in matrix form and solve them by the method of reduction:
x + 2y + z = 8, 2x + 3y – z = 11, 3x – y – 2z = 5.
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उत्तर
The given equations can be written in the matrix form as:
`[(1,2,1),(2,3,–1),(3,–1,–2)] [(x),(y),(z)] = [(8),(11),(5)]`
By R2 – 2R1 and R3 – 3R1, we get,
`[(1,2,1),(0,-1,-3),(0,-7,-5)] [(x),(y),(z)] = [(8),(-5),(-19)]`
By R3 – 7R2, we get,
`[(1,2,1),(0,-1,-3),(0,0,16)] [(x),(y),(z)] = [(8),(-5),(16)]`
∴ `[(x + 2y + z),(0 - y - 3z),(0 + 0 + 16z)] = [(8),(-5),(16)]`
By equality of matrices,
x + 2y + z = 8 ...(1)
– y – 3z = –5 ....(2)
16z = 16 ...(3)
From (3), z = 1
Substituting z = 1 in (2), we get,
–y – 3z = – 5,
–y – 3(1) = – 5,
–y = –2
∴ y = 2
Substituting y = 2, z = 1 in (1), we get,
x + 2y + z = 8
x + 2(2) + 1 = 8
x + 4 + 1 = 8
∴ x = 3
Hence, x = 3, y = 2, z = 1 is the required solution.
Notes
The second equation is modified as per the answer given in the textbook.
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