(x+yi)+(4+9i)=9-4i

Jasonbradley

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Jun 17, 2024

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Solving Complex Number Equations: (x + yi) + (4 + 9i) = 9 - 4i

This article will guide you through solving the complex number equation (x + yi) + (4 + 9i) = 9 - 4i. We'll use the properties of complex numbers to isolate the variables x and y.

Understanding Complex Numbers

Complex numbers are numbers of the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as the square root of -1 (i.e., i² = -1).

Key Properties:

• Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
• Equality: Two complex numbers are equal if and only if their real parts and imaginary parts are equal.

Solving the Equation

1. Simplify the left-hand side: (x + yi) + (4 + 9i) = (x + 4) + (y + 9)i

2. Equate real and imaginary parts: We now have: (x + 4) + (y + 9)i = 9 - 4i. This implies:

   • x + 4 = 9 (equating real parts)
   • y + 9 = -4 (equating imaginary parts)

3. Solve for x and y:

   • x = 9 - 4 = 5
   • y = -4 - 9 = -13

Solution

Therefore, the solution to the equation (x + yi) + (4 + 9i) = 9 - 4i is x = 5 and y = -13.

This means the complex number (x + yi) is 5 - 13i.