(x+2y)+i(2x-3y)=5-4i

Jasonbradley

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Jun 16, 2024

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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:

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

• Elimination Method:

  1. 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
  2. Add the two equations together: 7y = 14 => y = 2
  3. 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.