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Question
Express the following equations in matrix form and solve them by the method of reduction:
x + 3y + 2z = 6,
3x − 2y + 5z = 5,
2x − 3y + 6z = 7
Solution
Given equations are
x + 3y + 2z = 6
3x − 2y + 5z = 5
2x − 3y + 6z = 7
The matrix form is
`[(1, 3, 2),(3, -2, 5),(2, -3, 6)][(x),(y),(z)] = [(6),(5),(7)]`
Using `R_2 -> R_2 - 3R_1, R_3 -> R_3 - 2R_1`
`[(1, 3, 2),(0, -11, -1),(0, -9, 2)][(x),(y),(z)] = [(6),(-13), (-5)]`
Using `R_2 -> -1/11 R_2`
`[(1, 3, 2),(0, 1, 1/11),(0, -9, 2)][(x),(y),(z)] = [(6),(13/11), (-5)]`
Using `R_3 -> R_3 + 9R_2`
`[(1, 3, 2),(0, 1, 1/11),(0, 0, 31/11)][(x),(y),(z)] = [(6),(13/11),(62/11)]`
Putting the original equations for m in writing as
x + 3y + 2z = 6 ...(1)
`y + 1/11z = 13/11` ...(2)
`31/11z = 62/11` ...(3)
From (3): z = 2
From (2): `y + 2/11 = 13/11`
∴ y = 1
From (1): x + 3 + 4 = 6
∴ x = – 1
∴ x = – 1, y = 1, z = 2
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