Solving Complex Equations: (x + 2y) + i(2x  3y) = 5  4i
This article will guide you through solving the complex equation (x + 2y) + i(2x  3y) = 5  4i.
Understanding Complex Numbers
Before diving into the solution, let's briefly recap 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 (i² = 1).
Solving the Equation
To solve the equation (x + 2y) + i(2x  3y) = 5  4i, we need to equate the real and imaginary components on both sides of the equation:
 Real component: x + 2y = 5
 Imaginary component: 2x  3y = 4
Now we have a system of two linear equations with two unknowns (x and y). We can solve this system using various methods, such as:

Substitution Method:
 Solve one equation for one variable (e.g., solve the first equation for x: x = 5  2y)
 Substitute this expression into the second equation: 2(5  2y)  3y = 4
 Solve for y: 10  4y  3y = 4 => 7y = 14 => y = 2
 Substitute the value of y back into either of the original equations to find x: x + 2(2) = 5 => x = 1

Elimination Method:
 Multiply the first equation by 2 and the second equation by 1: 2(x + 2y) = 2(5) => 2x + 4y = 10 1(2x  3y) = 1(4) => 2x + 3y = 4
 Add the two equations together: 7y = 14 => y = 2
 Substitute the value of y back into either of the original equations to find x: x + 2(2) = 5 => x = 1
Solution
Therefore, the solution to the equation (x + 2y) + i(2x  3y) = 5  4i is x = 1 and y = 2.