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Question
Express the following equations in matrix form and solve them by the method of reduction:
`x + y = 1, y + z = 5/3, z + x 4/33`.
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Solution
The given equations can be written in the matrix form as:
`[(1,1,0),(0,1,1),(1,0,1)] [("x"),("y"),("z")] = [(1),(5/3),(4/3)]`
By R3 - R1, we get,
`[(1,1,0),(0,1,1),(0,-1,1)] [("x"),("y"),("z")] = [(1),(5/3),(1/3)]`
By R3 + R2, we get,
`[(1,1,0),(0,1,1),(0,0,2)] [("x"),("y"),("z")] = [(1),(5/3),(2)]`
∴ `[("x" + "y" + 0),(0+"y" + "z"),(0 + 0 + 2"z")] = [(1),(5/3),(2)]`
By equality of matrices,
x + y = 1 ...(1)
y + z = `5/3` .....(2)
2z = 2 ....(3)
From (3), z = 1
Substituting z = 1 in (2), we get,
y + 1 = `5/3`
`y = 5/3 - 1`
∴ y = `2/3`
Substituting y = 2/3 in (1), we get,
∴ x + y = 1
`x + 2/3 = 1`
`x = 1 - 2/3 = 1/3`
∴ `x = 1/3`
Hence, x = `1/3`, y = `2/3`, z = 1 is the required solution.
Notes
[Note: Question in the textbook is incomplete.]
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