(x+iy)(5+6i)=2+3i

3 min read Jun 17, 2024

Solving Complex Equations: (x + iy)(5 + 6i) = 2 + 3i

This article will guide you through the process of solving the complex equation (x + iy)(5 + 6i) = 2 + 3i for the unknown real numbers x and y.

Understanding Complex Numbers

A complex number is a number 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.

Expanding the Equation

First, we expand the left side of the equation by using the distributive property:

(x + iy)(5 + 6i) = 5x + 6xi + 5iy + 6iy²

Since i² = -1, we can simplify:

5x + 6xi + 5iy - 6y = 2 + 3i

Equating Real and Imaginary Components

We now have a single complex number on the left side and another on the right. For two complex numbers to be equal, their real and imaginary components must be equal.

This gives us two separate equations:

Real components: 5x - 6y = 2
Imaginary components: 6x + 5y = 3

Solving the System of Equations

We now have a system of two linear equations with two unknowns. There are several ways to solve this system, including:

Substitution Method: Solve one equation for one variable and substitute it into the other equation.
Elimination Method: Multiply one or both equations by constants to make the coefficients of one variable opposites, then add the equations together to eliminate that variable.

Let's use the elimination method. Multiply the first equation by 5 and the second equation by 6:

25x - 30y = 10
36x + 30y = 18

Now add the two equations together:

61x = 28

Therefore, x = 28/61.

Substitute this value of x back into either of the original equations to solve for y. Let's use the first equation:

5(28/61) - 6y = 2

Simplifying, we get:

y = -14/61

Solution

Therefore, the solution to the equation (x + iy)(5 + 6i) = 2 + 3i is:

x = 28/61
y = -14/61

This means the complex number (28/61 - 14/61i) is the solution to the equation.