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Question
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.
Solution
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" + 2"y" + "z"),(0 - "y" - 3"z"),(0 + 0 + 16"z")] = [(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 - 3 = - 5,
∴ y = 2
Substituting y = 2, z = 1 in (1), we get,
x + 4 + 1 = 8
∴ x = 3
Hence, x = 3, y = 2, z = 1 is the required solution.
Notes
[Note: Second equation is modified as per the answer given in the textbook.]
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