Question

Solve the following equation for x, y ∈ R:

(x + iy)(5 + 6i) = 2 + 3i

Answer 1

(x + iy)(5 + 6i) = 2 + 3i

∴ x + iy = `(2 + 3"i")/(5 + 6"i")`

∴ x + iy = `((2 + 3"i")(5 - 6"i"))/((5 + 6"i")(5 - 6"i")`

= `(10 - 12"i" + 15"i" - 18"i"^2)/(25 - 36"i"^2)`

= `(10 + 3"i" - 18(-1))/(25 - 36(-1)`   ......[∵ i² = – 1]

∴ x + iy = `(28 + 3"i")/61 = 28/61 + 3/61"i"`

Equating real and imaginary parts, we get

x = `28/61` and y = `3/61`.