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

Jasonbradley

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Jun 16, 2024

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Solving Complex Equations: A Step-by-Step Guide for (2i+3)x + (2-3i)y = 5-i

This article will guide you through solving the complex equation (2i+3)x + (2-3i)y = 5-i. We will break down the steps involved in finding the values of x and y that satisfy the equation.

Understanding Complex Equations

A complex equation involves complex numbers, which are numbers of the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit (i² = -1). To solve a complex equation, we need to find the values of the unknown variables that make the equation true.

Solving the Equation

1. Separate the Real and Imaginary Parts: Since the equation involves complex numbers, we need to separate the real and imaginary parts. We can do this by expanding the equation:

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

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

2. Equate Real and Imaginary Parts: For the equation to be true, the real parts on both sides must be equal, and the imaginary parts must also be equal. This gives us two separate equations:

   • Real part: 3x + 2y = 5
   • Imaginary part: 2x - 3y = -1

3. Solve the System of Equations: Now we have two linear equations with two unknowns. We can solve this system using various methods like substitution, elimination, or matrix methods.

   Using elimination:

   • Multiply the first equation by 3 and the second equation by 2 to make the coefficients of 'y' opposites:
     • 9x + 6y = 15
     • 4x - 6y = -2
   • Add the two equations together to eliminate 'y':
     • 13x = 13
   • Solve for 'x':
     • x = 1
   • Substitute the value of 'x' back into either of the original equations to solve for 'y':
     • 3(1) + 2y = 5
     • 2y = 2
     • y = 1

Solution

Therefore, the solution to the complex equation (2i+3)x + (2-3i)y = 5-i is:

• x = 1
• y = 1

This means that substituting these values into the original equation will make the equation true.