Question

Solve the following equations for real x and y.
(2-3i)(x+yi) = 4+i

Answer 1

see explanation

Explanation:

Expand the left side and equate to like terms on the right side

(2 - 3i)(x + yi)

= 2x + 2iy - 3ix - 3yi² ← using i² = - 1, then

= 2x + 2iy - 3ix + 3y → compare with 4 + i

2x + 3y = 4 → (1)

- 3x + 2y = 1 → (2)

Multiply (1) by 3 and (2) by 2

6x + 9y = 12 → (3)

- 6x + 4y = 2 → (4)

Add (3) and (4) term by term to eliminate term in x

13y = 14 ( divide both sides by 13 )

y =
`(14)/(13)`

Substitute this value into (1) and solve for x

2x +
`(42)/(13)` =
`(52)/(13)`

Subtract
`(42)/(13)` from both sides

2x =
`(10)/(13)` ← divide both sides by 2

x =
`(5)/(13)`

Hence x =
`(5)/(13)` and y =
`(14)/(13)`