Question

If \( (3i + 1)x + (4i + 4)y + 5 = 0 \) with \( x, y \) being real and \( i = \sqrt{-1} \), then x = _________.

Hint:

For equations involving complex numbers, separate the real and imaginary parts and solve the system of equations.

Answer 1

Correct Answer: 2.4

The given equation is: \[ (3i + 1)x + (4i + 4)y + 5 = 0 \] Separate the real and imaginary parts: \[ (3i)x + x + (4i)y + 4y + 5 = 0 \] Group real and imaginary terms: \[ (x + 4y + 5) + i(3x + 4y) = 0 \] For the equation to hold, both the real and imaginary parts must be equal to zero. So: \[ x + 4y + 5 = 0 \quad \text{(1)} \] \[ 3x + 4y = 0 \quad \text{(2)} \] From equation (2), solve for \( y \): \[ y = -\frac{3}{4}x \] Substitute this into equation (1): \[ x + 4\left(-\frac{3}{4}x\right) + 5 = 0 \] \[ x - 3x + 5 = 0 \] \[ -2x + 5 = 0 \] \[ x = \frac{5}{2} = 2.5 \] Thus, \( x = \boxed{2.5} \).