(x+6i)=(3-i)+(4-2yi)

Jasonbradley

2 min read

·

Jun 17, 2024

20

Share

Solving Complex Number Equations: (x + 6i) = (3 - i) + (4 - 2yi)

This article will guide you through solving the complex number equation (x + 6i) = (3 - i) + (4 - 2yi). We will use the properties of complex numbers to isolate the variables x and y.

Understanding Complex Numbers

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1.

Key Properties:

• Equality: Two complex numbers are equal if and only if their real and imaginary parts are equal.
• Addition and Subtraction: Complex numbers are added and subtracted by adding or subtracting their corresponding real and imaginary parts.

Solving the Equation

1. Simplify the right-hand side:

   (3 - i) + (4 - 2yi) = (3 + 4) + (-1 - 2y)i = 7 - (1 + 2y)i

2. Equate real and imaginary parts:

   Since the left-hand side is x + 6i and the right-hand side is 7 - (1 + 2y)i, we can equate their real and imaginary parts:

   • Real parts: x = 7
   • Imaginary parts: 6 = -(1 + 2y)

3. Solve for y:

   • 6 = -1 - 2y
   • 2y = -7
   • y = -7/2

Solution

Therefore, the solution to the equation (x + 6i) = (3 - i) + (4 - 2yi) is:

• x = 7
• y = -7/2

This means the equation is satisfied when we substitute these values for x and y.