Question

Solve the following equation for x, y ∈ R:

2x + i⁹ y (2 + i) = x i⁷ + 10 i¹⁶ 

Answer 1

2x + i⁹ y (2 + i) = x i⁷ + 10 i¹⁶ 
∴ 2x + (i⁴)².i.y (2 + i) = x (i²)³.i + 10.(i⁴)⁴
∴ 2x + (1)².iy (2 + i) = x (– 1)³.i +10 (1)⁴    ...[∵ i² = – 1, i⁴ = 1]
∴  2x + 2yi + yi² = – xi + 10
∴ 2x + 2yi – y + xi = 10
∴ (2x – y) + (x + 2y)i = 10 + 0.i
Equating real and imaginary parts, we get
2x – y = 10          ...(i)
and x + 2y = 0    ...(ii)
Equation (i) x 2 + equation (ii) gives
5x = 20
∴ x = 4
Putting x = 4 in (i), we get
2(4) –  y = 10
∴ y = 8 – 10
∴ y = – 2
∴ x = 4 and y = – 2