Solving Complex Equation with Real Variables x and y

Published on September 6, 2026

Algebra

Find real numbers x and y that satisfy the complex number equation 2x - 3i + yi - |3 - 4i| = y + xi by equating the real and imaginary parts.

Question

Answer

First, calculate the modulus of the complex number :

Substitute this value back into the original equation:

Group the real and imaginary parts on both sides of the equation:

For two complex numbers to be equal, their real parts must be equal and their imaginary parts must be equal. This gives us a system of linear equations:

Solve the system by substituting (from the second equation) into the first equation:

Find using the value of :

Final answer:

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