(2+3i)x+(3-2i)y=8-i

Jasonbradley

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Jun 16, 2024

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Solving Complex Equations: A Step-by-Step Guide

This article will guide you through solving the complex equation (2+3i)x + (3-2i)y = 8-i. We'll break down the process into manageable steps to ensure a clear understanding.

Understanding Complex Numbers

Before we dive into solving, let's briefly recap complex numbers. They 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.

Solving the Equation

1. Separate Real and Imaginary Parts:

   Our equation involves complex numbers multiplied by variables. To solve for x and y, we need to separate the real and imaginary parts:

   (2+3i)x + (3-2i)y = 8-i

   Expanding the left side:

   (2x + 3ix) + (3y - 2iy) = 8 - i

   Combining real and imaginary terms:

   (2x + 3y) + (3x - 2y)i = 8 - i

2. Equate Real and Imaginary Components:

   For the equation to hold true, the real parts on both sides must be equal, and the imaginary parts on both sides must be equal. This gives us two separate equations:

   • Real part: 2x + 3y = 8
   • Imaginary part: 3x - 2y = -1

3. Solve the System of Equations:

   We now have a system of two linear equations with two unknowns. We can solve this system using various methods, such as substitution or elimination. Let's use elimination:

   • Multiply the first equation by 2: 4x + 6y = 16
   • Multiply the second equation by 3: 9x - 6y = -3
   • Add the two equations: 13x = 13
   • Solve for x: x = 1

   Now substitute the value of x (1) back into either of the original equations. Let's use the first equation:

   • 2(1) + 3y = 8
   • 3y = 6
   • Solve for y: y = 2

Solution

Therefore, the solution to the complex equation (2+3i)x + (3-2i)y = 8-i is x = 1 and y = 2.

This solution can be verified by substituting the values of x and y back into the original equation.