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Question
Solve the following equation for x, y ∈ R:
2x + i9y (2 + i) = xi7 + 10i16
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Solution
2x + i9y (2 + i) = xi7 +10i16
i9 = i8 x i = (i2)4i = (– 1)4i = i
i7 = i6 x i = (i2)3i = (–1)3i = – i
i16 = (i2)8 = (– 1)8 = 1
∴ given equation becomes
2x + iy(2 + i) = – xi + 10
∴ 2x + 2iy + i2y = – xi + 10
∴ 2x + 2iy – y = – xi + 10 ...[∵ i2 = – 1]
∴ 2x + 2iy – y + xi – 10 = 0
∴ (2x – y – 10) + (x + 2y)i = 0
∴ (2x – y) + (x + 2y)i = 10 + 0i
Equating the real and imaginary parts separately, we get,
2x – y = 10 ...(1)
and x + 2y = 0 ...(2)
Multiplying equation (1) by 2, we get, 4x – 2y = 20
Adding this equation with equation (2), we get,
5x = 20
∴ x = 4
∴ from (2), 4 + 2y = 0
∴ 2y = – 4
∴ y = – 2
Hence, x = 4, y = – 2.
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