(3-2i)z=-5+4i

Jasonbradley

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Jun 16, 2024

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Solving Complex Equation: (3-2i)z = -5 + 4i

This article will guide you through solving the complex equation (3-2i)z = -5 + 4i for the complex variable z.

Understanding Complex Numbers

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1.

Important properties:

• Addition/Subtraction: Complex numbers are added and subtracted by adding/subtracting their real and imaginary components separately.
• Multiplication: Multiply complex numbers as you would binomials, remembering that i² = -1.
• Division: To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

Solving the Equation

1. Isolate z: To solve for z, we need to divide both sides of the equation by (3-2i):

   z = (-5 + 4i) / (3 - 2i)
   

2. Multiply by the Conjugate: The conjugate of (3-2i) is (3+2i). Multiplying both the numerator and denominator by this conjugate:

   z = [(-5 + 4i) * (3 + 2i)] / [(3 - 2i) * (3 + 2i)]
   

3. Expand and Simplify: Expanding the products in the numerator and denominator, using the distributive property and remembering i² = -1:

   z = (-15 - 8 + 12i + 8i) / (9 + 4) 
   z = (-23 + 20i) / 13
   

4. Express in Standard Form: To express the solution in standard form (a + bi):

   z = -23/13 + (20/13)i 
   

Solution

Therefore, the solution to the complex equation (3-2i)z = -5 + 4i is z = -23/13 + (20/13)i.