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)

Answer 1

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