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Question
If x + y + z = 3, x + 2y + 3z = 4, x + 4y + 9z = 6, then (y, z) = _______
Options
(–1, 0)
(1, 0)
(1, –1)
(–1, 1)
Solution
If x + y + z = 3, x + 2y + 3z = 4, x + 4y + 9z = 6, then (y, z) = (1, 0).
Explanation:
Matrix form of equations is
`[(1, 1, 1), (1, 2, 3), (1, 4, 9)] [(x), (y), (z)] = [(3), (4), (6)]`
R2 − R1, R3 − R1
`[(1, 1, 1), (0, 1, 2), (0, 3, 8)] [(x), (y), (z)] = [(3), (1), (3)]`
R3 − 3R2
`[(1, 1, 1), (0, 1, 2), (0, 0, 2)] [(x), (y), (z)] = [(3), (1), (0)]`
`[(x + y + z), (y + 2z), (2z)] = [(3), (1), (0)]`
By equality of matrices, we get
x + y + z = 3 ...(1)
y + 2z = 1 ...(2)
2z = 0 ...(3)
∴ z = 0
Putting z = 0 in (2),
y + 0 = 1
∴ y = 1
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