Question

Solve the following equation for x, y ∈ R:

2x + i⁹y (2 + i) = xi⁷ + 10i¹⁶

Answer 1

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.