(x+yi)+(8-5i)=4+3i

Jasonbradley

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Jun 17, 2024

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Solving Complex Number Equations: (x + yi) + (8 - 5i) = 4 + 3i

This article will guide you through solving the complex number equation (x + yi) + (8 - 5i) = 4 + 3i.

Understanding Complex Numbers

Complex numbers are numbers 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.

Solving the Equation

1. Combine real and imaginary terms:

   • The real terms are x and 8, while the imaginary terms are yi and -5i.

   • Combine the real terms on one side and the imaginary terms on the other:

     (x + 8) + (yi - 5i) = 4 + 3i

2. Equate real and imaginary components:

   • Since the equation holds true for all values of x and y, we can equate the real and imaginary components on both sides.

   • This gives us two separate equations:

     x + 8 = 4 yi - 5i = 3i

3. Solve for x and y:

   • Solve the first equation for x:

     x = 4 - 8 = -4

   • Solve the second equation for y:

     yi = 3i + 5i = 8i y = 8 (since i cancels out on both sides)

Solution

Therefore, the solution to the equation (x + yi) + (8 - 5i) = 4 + 3i is x = -4 and y = 8.

This can be expressed in the complex number form as -4 + 8i.